Cone-equivalent nilpotent groups with different Dehn functions

For every $k\geqslant 3$, we exhibit a simply connected $k$-nilpotent Lie group $N_k$ whose Dehn function behaves like $n^k$, while the Dehn function of its associated Carnot graded group $\mathsf{gr}(N_k)$ behaves like $n^{k+1}$. This property and its consequences allow us to reveal three new phenomena. First, since those groups have uniform lattices, this provides the first examples of pairs of finitely presented groups with bilipschitz asymptotic cones but with different Dehn functions. The second surprising feature of these groups is that for every even integer $k \geqslant 4$ the centralized Dehn function of $N_k$ behaves like $n^{k-1}$ and so has a different exponent than the Dehn function. This answers a question of Young. Finally, we turn our attention to sublinear bilipschitz equivalences (SBE). Introduced by Cornulier, these are maps between metric spaces inducing bi-Lipschitz homeomorphisms between their asymptotic cones. These can be seen as weakenings of quasiisometries where the additive error is replaced by a sublinearly growing function $v$. We show that a $v$-SBE between $N_k$ and $\mathsf{gr}(N_k)$ must satisfy $v(n)\succcurlyeq n^{1/(2k + 2)}$, strengthening the fact that those two groups are not quasiisometric. This is the first instance where an explicit lower bound is provided for a pair of SBE groups.


INTRODUCTION
The goal of this work is to improve our understanding of the large-scale geometry of simply connected nilpotent Lie groups and, more specifically, of an asymptotic invariant called the Dehn function, which encodes fundamental geometric and algebraic information on the group. Given a simply connected Lie group equipped with a left-invariant Riemannian metric, the Dehn function ( ) is the smallest real number such that every rectifiable loop of length ⩽ in admits a filling by a Lipschitz disk of area ⩽ ( ). An important feature of the Dehn function is the invariance of its asymptotics under quasi-isometry (see Subsection 3.1). The study of filling functions of nilpotent Lie groups is a very difficult subject that has been deeply explored by Gromov, who initiated it [23,24], and other authors (for example, [1,21,34,40,41,43]). The main result of this paper should be seen as a contribution to this important subject. However, one of its key applications and the choice of groups studied can be better appreciated in the wider context of the study of the large-scale geometry of simply connected nilpotent groups. We will thus start by recalling known facts and central open problems in this area. A reader who directly wants to proceed to our results can go straight to Subsection 1.2.

Background on the large-scale geometry of nilpotent groups
A motivation for focussing on simply connected Lie groups rather than discrete groups is that every finitely generated nilpotent group maps with finite kernel onto a lattice in a unique simply connected nilpotent Lie group (called its real Malcev completion) [27]. It follows that the quasiisometry classification of finitely generated nilpotent groups reduces to that of simply connected nilpotent Lie groups, which is conjectured to have the following very neat formulation (see [12,Conjecture 19.114]).

Conjecture 1.1. Two simply connected nilpotent Lie groups are quasi-isometric if and only if they are isomorphic.
Conjecture 1.1 is more commonly stated in the discrete case: two finitely generated torsion-free nilpotent groups are quasi-isometric if and only if they have isomorphic real Malcev completions (this is mentioned as an open question in [19]). It is tempting to ask whether a quasi-isometry between two such groups implies that they are commensurable. This turns out to be false. Indeed, for any ring , let ℍ ( ) denote the -dimensional Heisenberg group over that ring. Then ℍ ( [ √ 2]) and ℍ ( ) 2 are both (uniform) lattices in ℍ ( ) 2 , therefore they are quasi-isometric. However, their rational Malcev completions are not isomorphic, which is equivalent to saying that the groups are not commensurable [27].
The lowest dimensional example of a pair of simply connected nilpotent Lie groups for which Conjecture 1.1 is still open occurs in dimension 5 (for a complete overview of the state of the art in dimension ⩽ 6 we refer to [12]). This shows that we are still far from having a complete proof even in low dimensions. On the other hand, there is ample evidence pointing towards the veracity of Conjecture 1.1, with one of the first striking results being Pansu's theorem. To state it precisely we need to recall the notions of a Carnot graded Lie algebra (respectively, a Carnot graded nilpotent Lie group).
We denote 1 = , +1 = [ , ] the lower central series of the Lie algebra (respectively, the lower central series of the group ). A Lie algebra (respectively, group ) has step † if is the smallest integer such that +1 = {0} (respectively, +1 = {1}). The lower central series gives rise to a filtration of in the sense that [ , ] ⊂ + . A Lie algebra is called Carnot gradable if this filtration comes from a grading, that is, a decomposition = ⨁ satisfying = ⨁ ⩾ and [ , ] ⊂ + ; such a grading is called a Carnot grading. It is always possible to associate a Carnot graded Lie algebra ( ) to any nilpotent Lie algebra by letting ( ) = ⨁ ⩾1 for = ∕ +1 and defining the Lie bracket in the obvious way to make a grading (see Subsection 7.2 for more details). We denote ( ) the simply connected nilpotent Lie group whose Lie algebra is ( ). The pair ( ( ), 1 ) is then called a Carnot-graded group (some authors say stratified group). Observe that ( ) has the same dimension and step as .
We say for convenience that two groups are cone equivalent if their asymptotic cones with respect to any given non-principal ultrafilter are bi-Lipschitz ‡ . It is easy to see that two groups that are quasi-isometric are cone equivalent. Pansu's fundamental theorem provides a complete classification of simply connected nilpotent groups up to cone equivalence. the importance of understanding nilpotent Lie groups that are not Carnot gradable, even if the tools at hand are much more limited.
One way of obtaining interesting examples of nilpotent Lie groups that are not Carnot gradable is a general construction called a central product. Given two Lie algebras and , central subspaces ⊂ and ′ ⊂ , and an isomorphism ∶ → ′ , we define the central product = × to be the quotient of the direct product × by the central ideal {( , − ( )); ∈ }.
Let (respectively, ) be the maximal integer such that (respectively, ′ ) is contained in the th (respectively, th) term of the lower central series of (respectively, ). If > ⩾ 2 and and are Carnot graded with 1-dimensional centres, it is easy to check that the Lie algebra is not Carnot gradable and that ( ) is isomorphic to the direct product × ( ∕ ′ ).
To introduce the explicit family of groups that will form our main object of study, we start by recalling a classical class of Carnot graded Lie algebras. We denote by the corresponding simply connected Lie group. The semi-direct product Λ ≅ −1 ⋊ , with ( 1 )( ) = +1 , 2 ⩽ ⩽ − 1, and ( 1 )( ) = 0, defines a lattice in , where we denote by 1 the generator of and by 2 , … , the generators of −1 . This provides us with a natural presentation (Λ ) of Λ which we will use later. † If ⩾ 3, the centre of is the 1-dimensional subalgebra spanned by ∶= . For 3 ⩽ ⩽ we define the Lie algebra , to be the central product (defined unambiguously) of and . We let , be the corresponding simply connected Lie group. , admits a uniform lattice Γ , which is simply the central product of Λ and Λ . As a concrete example, observe that 3,3 and Γ 3,3 are the 5-dimensional Heisenberg group ℍ 5 ( ) and its integer lattice ℍ 5 ( ), respectively.
The groups , and their corresponding Lie algebras , will form our main object of study in this paper; in particular the cases when = − 1 or = . A key motivation for this is that the Lie algebras , for , ⩾ 3 are Carnot gradable if and only if = and thus that the corresponding groups , are not isomorphic to their asymptotic cones if ≠ . Indeed, for 2 < < , the associated Carnot-graded Lie algebra ( , ) is isomorphic to the direct product × −1 (note that 2 = 2 ) and thus ( , ) ≅ × −1 . Moreover, we observe that , and thus ( , ) are max( − 1, − 1)-step nilpotent. We will now proceed to exploit the difference between , and × −1 to reveal the first phenomenon from the abstract.

A family of pairs of cone equivalent groups with different Dehn functions
We shall use the following notation ‡ : if , g are functions defined on ⩾0 we write ( ) ≼ g( ) if | ( )| ⩽ |g( + )| + for some ⩾ 0, and ( ) ≍ g( ) if ( ) ≼ g( ) ≼ ( ). Finally, ( ) ≺ g( ) means that ( ) ≼ g( ) holds but ( ) ≍ g( ) does not. † Note that using the same notation for the generators of the lattice Λ and the generators of the Lie algebra will not cause any confusion, as it will always be clear from context which one of the two we are working with. ‡ We emphasise that in contrast to a common convention in the setting of Dehn functions we do not allow for a linear term in the definition of ≼. This has two reasons: (i) we do not consider any Dehn functions of hyperbolic groups, and (ii) we require this stronger form of equivalence in the context of SBE below.
On the other hand, it follows from classical arguments that ( , −1 ) ≅ × −2 has Dehn function ≍ . Hence, we deduce from Pansu's theorem the following corollary.
Corollary B. For every ⩾ 3, there is a pair of finitely generated (or simply connected Lie) -nilpotent groups with bi-Lipschitz asymptotic cones but whose Dehn functions have different growth types.
Note that for = 3 Theorem A does not hold in the − 1 case: while the Dehn function of 3,3 = ℍ 5 ( ) is known to be quadratic [1,28], the Dehn function of 3,2 ≅ ℍ 3 ( ) × is cubic. We also emphasise that the fact that 3,3 does have quadratic Dehn function will later form the basis for our induction argument in the proof of the upper bounds in Theorem A.
It has been known since Gromov [23] that topological properties of asymptotic cones impose restrictions on Dehn functions (for example, if the asymptotic cone is a real tree, respectively, simply connected, then the Dehn function is linear, respectively, polynomially bounded). A consequence of a theorem of Papasoglu is that if ( ) ≲ , then ( ) ≲ + for every > 0 [17, 2.7; 32]. Corollary B shows that there is no converse to this theorem, proving that the fine behaviour of the Dehn function is not always captured by the asymptotic cone. The fact that central products can have a lower Dehn function than their factors has been noticed by Olshanskii and Sapir [28], and by Young [43] for a large class of examples. However, in all situations studied by these authors, the groups in question are actually step 2 nilpotent and therefore Carnot gradable.
The lowest dimensional occurrence of the phenomenon described by Corollary B is in dimension 6. Indeed, the group 4,3 shares its asymptotic cone with two other 6-dimensional step 3 groups and the Dehn function of 4,3 is cubic, whereas for the two others it is quartic. We refer to Section 10 for a detailed discussion of all 6-dimensional nilpotent Lie algebras and the Dehn functions of their associated simply connected nilpotent Lie groups.

Sublinear bi-Lipschitz equivalence
Considering the Dehn function of the group 4,3 was suggested by Cornulier and triggered our work [11,Question 6.20]. Cornulier's motivation is coming from the study of SBEs between nilpotent groups. He proves that for every pair ( , ( )) where has step one can choose of the form ( ) ≍ with = 1 − 1∕ [11,Theorem 1.21]. The Dehn function is well-known to be invariant under quasi-isometry and Cornulier observed a weaker stability result for the Dehn function under SBE: having Dehn functions with different exponents implies an asymptotic lower bound ≽ for some > 0 on the possible functions such that there can exist an ( )-SBE. With this in mind he suggested the pair ( 4,3 , 4 × 2 ) as a possible example satisfying such a lower bound [11,Example 6.19]. We confirm Cornulier's intuition and, more generally, prove the following result.
Actually the precise exponent in Theorem C is deduced from a slightly stronger version of Theorem A, saying that the filling occurs in a ball of radius comparable to the length of the loop † .
This lower bound should be compared with the asymptotic upper bound of −2 −1 following from Cornulier's general estimates. It would be interesting to understand the precise asymptotics of the exponent as a function of as → +∞: in particular, does it tend to zero? Remark 1.5. As already mentioned, the fact that the groups considered in Theorem C (or their lattices) are not quasi-isometric is also a consequence of [37,Theorem 1.2]. Indeed we shall see that 2 (Λ × Λ −2 ) = 2 (Γ , −1 ) + , where is 1 or 2 according to whether is even or odd (see Lemma 7.12).

Centralised and regular Dehn functions differ for nilpotent groups
We now recall the algebraic definition of the Dehn function. Given a presentation (not necessarily finite) ⟨ | ⟩ of a group , one can define its Dehn function as follows: we call an element of the free group generated by null-homotopic (in ) if it represents the trivial element in . For every null-homotopic word ∈ we define Area( ) to be the minimal integer such that where ∈ and ∈ ±1 . The Dehn function , , ( ) is the (possibly infinite) infimum of Area( ) over all null-homotopic words ∈ of length at most . If the group is finitely presented, then the Dehn function takes finite values and its asymptotic behaviour does not depend on the choice of finite presentation. A similar statement holds for compactly presented groups (see Section 3).
In [3], Baumslag, Miller and Short introduce the closely related notion of centralised Dehn function of a presentation ⟨ | ⟩ of a group , which they define as follows: Definition 1.6. Denote ℜ the normal subgroup of generated by . Given a null-homotopic word ∈ , we define its central area Area cent ( ) to be the minimal integer such that = ∏ =1 in ℜ∕[ , ℜ], with ∈ ±1 . The centralised Dehn function cent , , ( ) is the (possibly infinite) infimum of Area cent ( ) over all null-homotopic words of length at most .
As for the Dehn function, one can show that the asymptotic behaviour of the centralised Dehn function of a finitely presented group does not depend on a specific choice of finite presentation, so we simply denote it by cent . Note that we have cent ⩽ by definition. It turns out that cent is in general easier to estimate as it is closely related to the second cohomology group of , or, equivalently, to the central extensions of . In particular, we have the following useful characterisation of the centralised Dehn function for torsion-free nilpotent groups. Proposition 1.7 (see Proposition 7.2). Let Γ be a torsion-free nilpotent group and let be the Lie algebra of its Malcev completion. Then cent Γ ( ) ≍ , where is the largest integer such that admits a central extension →̃→ whose kernel belongs tõ.
Such a central extension will be called -central in the sequel. The centralised Dehn function was used in [3] to obtain sharp lower bounds on the Dehn functions of certain nilpotent groups. In [41], Young mentions that for nilpotent groups it is unknown whether cent Γ ( ) ≍ Γ ( ). Later Wenger exhibited a 2-step nilpotent group whose Dehn function strictly lies between quadratic and 2 log [40], therefore answering Young's question negatively. Here we show that even the growth exponents of the two functions can be different. Hence, the Dehn function and the centralised Dehn function have different exponents for Γ 2 +1,2 .

Structure of the paper
In Section 2, we give an overview of the proof of our main results. In Section 3, we introduce basic notions and results regarding compact presentations, Dehn functions and filling diameters. In Section 4, we prove the upper bound in Theorem A for = 4 as a warm-up for the general case. In Section 5, we set the stage for the proof of the upper bound in Theorem A for general , by deriving an explicit compact presentation for , and then proving several preliminary results satisfied by words in its generators. Section 6 contains the proof of the upper bound in Theorem A. In Section 7, we explore the existence of central extensions of central products. In Section 8, we derive the lower bounds in Theorem A for all , showing that for odd the lower bounds on the Dehn function of , −1 provided by the centralised Dehn function are not optimal, thus also completing the proof of Theorem D. Section 9 is concerned with applying our results in the theory of SBEs, leading to a proof of Theorem C. In Section 10, we give an overview of the Dehn functions of nilpotent groups of dimension less or equal to six. Finally, we list some open questions and speculations arising from our work in Section 11.

Groups and Lie algebras
When working with words ( ) in the generators of a group with presentation  = ⟨ | ⟩ we will be careful to distinguish equalities of words and equalities of their corresponding elements in the group. To do so, for words 1 ( ) and 2 ( ) we will write 1 ( ) = 2 ( ) if they are equal as words and 1 ( ) ≡ 2 ( ) (with respect to  or ) if they represent the same element of the group. Whenever this is not clear from context we will make sure to mention the presentation (or group) explicitly when using ≡. We will write [ ] for the group element represented by a word if we want to explicitly distinguish it from the word. We will denote by ( ) the word length of a word ( ) and for a group element g ∈ by |g| ∶= Cay , (1, g) the distance of g from the origin in the Cayley graph.
We call a word ( ) central if [ ] is a central element of the group .

Asymptotic comparisons
We shall use the notation ≲ to mean that there exists some < ∞ only depending on such that ⩽ . Similarly, we denote ≃ if ≲ and ≲ . Sometimes we will also say is in ( ) if ≲ and = ( ) if ≃ .

OVERVIEW OF THE PROOF
To provide the reader with an intuition for the proofs in this paper we now briefly explain the moral ideas behind why the groups , −1 satisfy the conclusions of Theorem A and Theorem D.
The proof of Theorem A and Theorem D has three fundamental parts, which make up most of this paper.
(1) The proof of the upper bound of −1 on the Dehn functions of , −1 and , . This will make up by far the biggest part of this work and will be contained in Sections 4-6. (2) The proof that , −1 admits no ( − 1)-central extension when is odd, which will be contained in Section 7.
(3) The proof that the Dehn function of , −1 is nevertheless bounded below by −1 , irrespectively of the parity of , which will be contained in Section 8.
Parts (2) and (3) turn out to be easier to explain in the setting of Lie algebras, while we postpone most of the explanation of Part (1) to Subsection 2.2. So, we will adopt the Lie algebra point of view here.

The fundamental reason for why everything works
At the base of all three parts is the existence of the central element which connects the two factors of the central product via the identification = ( ) = in , = × , respectively, its group theoretic analogue. † To avoid confusion, let us mention that when we work with the Lie group , we will denote the generators of thefactor by 1 , − +2 , … , −1 , = , as this turns out to be more convenient, while for the Lie algebra setting the indices chosen here turn out to be easier to work with. The Lie algebra approach and thus this choice of indices will only appear in Section 7.
From the Lie algebra point of view this comes into play as follows: the differential of is = − 1 ∧ −1 in , and thus in × −2 = ( , −1 ), but The computational consequence is that it will be more difficult for a form ∈ ⋀ 2 * , to have vanishing exterior derivative if it has terms with a non-trivial contribution than is the case in ⋀ 2 * . Indeed, being a linear combination of two basis elements means that its differential 'interacts non-trivially' with more other basis elements than if it only had one summand.
From the group theory point of view the relation = = will enable us to move central words in the between factors, allowing us to commute them more easily with other words in the .
We briefly expand on how these observations come into play in Parts (1)-(3), thereby providing the moral idea of why and how our proof works.
Part (1): Let us just mention at this point that the argument is by induction on and ultimately boils down to the idea that we can commute central words of length in the with other words of length in the at cost −1 rather than (as one might naively expect). We achieve this by passing through the second factor of the central product via the subgroup −1, −1 ⩽ , −1 , which has Dehn function −2 by induction. Actually proving this for general will require a chain of combinatorial results. However, a good intuition for the general ideas should be attainable from the case = 4, which we will sketch in Subsection 2.2 and prove in detail in Section 4 Part (2): In the -factor of the Carnot Lie algebra × −1 = ( , −1 ) associated to , −1 there is a 2-form 2 ′ with ′ = ⌈ ∕2⌉, which defines a (2 ′ − 1)-central extension of and thus of × −2 . A precise definition of this form will be given in Subsection 7.4. Note that if is even (2 ′ − 1) = − 1, whereas if is odd 2 ′ − 1 = . Interestingly, the form 2 ′ only defines a cocycle in 2 ( , −1 , ) if is even, that is, its exterior derivative does not vanish when is odd. In terms of linear algebra the non-vanishing of its exterior derivative precisely boils down to the fact that has one summand more in , than in × −2 due to the central product structure.
Irrespectively of the parity of there are no other forms defining -central extensions for ⩾ − 1 in 2 ( , −1 , ) and we deduce that , −1 admits a ( − 1)-central extension if and only if is even. In combination with Theorem A this proves Theorem D Part (3): On first sight there is one more candidate for a cocycle defining a ( − 1)-central extension of , −1 , namely 1 ∧ −1 . But of course it is a non-candidate, because it is the 2-form defining the 'obvious' ( − 2)-central extension × −1 → , −1 .
However, this 'false candidate' for a ( − 1)-central extension is precisely the reason why the Dehn function of , −1 for odd is bigger than one might expect from the centralised Dehn function.
Indeed, as we already mentioned, 1 ∧ −1 defines a cocycle in 2 ( , −1 , ) and in a sense the only problem is that the commutator [ 1 , −1 ] ∈ −1 , −1 on which it is non-trivial is equal to and in particular does not vanish in , −1 .
We found a solution to overcome this issue and confirm our intuition that the Dehn function of , −1 is bounded below by −1 also when is odd. The idea is to exploit a 'perturbation' of the 2-form 1 ∧ −1 in order to show that the null-homotopic loops [ 1 , [ 1 , … , [ 1 , 2 ] … ]] have area bounded below by −1 . We will explain the technique used for this approach in the first half of Subsection 2.2.

Sketch of proof of Theorem A
The proof of Theorem A will cover the largest part of this paper. It splits into two independent parts: the proof of the lower bound, and the proof of the upper bound. The former will be contained in Section 8, while the latter will span Sections 4-6. To make it more accessible, we will provide a brief summary of the main ideas involved. We start by discussing the proof of the lower bound. When is even, then the lower bound is simply given by Theorem D. The case when is odd is much more involved and requires new ideas. Our method is inspired by Thurston's proof of the exponential lower bound on the Dehn function of the real 3-dimensional SOL group [18]. Thurston proceeds as follows: he exhibits a 1-form on such that is left-invariant, and a sequence of loops of length such that the integral of along is ⩾ for some > 1. A direct application of Stokes' theorem then implies that the area of any smooth embedded surface bounded by must be bounded below by for some constant > 0 only depending on and on a choice of left-invariant Riemannian metric on .
The first step in our argument consists of the observation that Thurston's assumption that is invariant can be relaxed to the weaker assumption that it is 'bounded'. To that purpose, we define the space of bounded -forms on to be the space of forms such that sup g∈ ‖(g * ) 1 where ‖ ⋅ ‖ is a norm on ⋀ * (note that the boundedness condition does not depend on a choice of such a norm). It is quite immediate to see that Thurston's approach works verbatim replacing the condition that is invariant by the condition that is bounded. We note that a related approach was developed by Gersten, who explains how ∞ -cocycles can be used to obtain lower bounds on the Dehn function of a finitely presented group [20, 2.7].
The second step and main innovation in our argument is the construction of a suitable bounded 2-form by 'deforming' a well-chosen invariant form. For this we exploit the central product structure of our groups. We start by observing that , −1 maps surjectively to −1 . We shall consider a 2-cocycle of −1 associated to its central extension and consider an invariant 2-form representing it in de Rham cohomology. We will then consider a relation of length in −1 and a primitive of whose integral along (a continuous path associated to) has size ≍ −1 . Although the word corresponding to would not define a relation in , −1 , its commutator [ , ] for a suitable word will. The problem at this point is that the integral of along [ , ] will be zero. So, we shall perform a suitable 'local perturbation' of , obtaining a 1-form ′ whose integral along [ , ] is ≍ −1 , and such that ′ while not being invariant anymore will remain bounded. This will show that the area of [ , ] in −1 (and a fortiori in , −1 ) is at least −1 .
Actually when trying to implement the previous argument, we run into a regularity problem: we have to deal with forms that are not smooth, preventing us from using Stokes' theorem. A solution would be to smoothen our forms so that the previous argument could be applied directly. However, this would make our computations more cumbersome. We chose instead to privilege an alternative approach, which better suits the study of Dehn functions associated to compact presentations. The idea is to replace the condition that is bounded by the fact that the integral of along any loop of bounded length is bounded. This condition is easy to work with and has the nice advantage of making sense for continuous 1-forms. Moreover, it satisfies a discrete version of Stokes' theorem, inspired by [13, section 12.A].
We now turn to the proof of the upper bound in Theorem A that occupies the largest part of the paper and is our main contribution to the subject. In Section 4, we start by proving the upper bound 4,3 ( ) ≲ 3 . Indeed, while containing the main idea, this bound turns out to be consider-ably easier to obtain than the more general bound , −1 ( ) ≲ −1 . At the end of Section 4, we shall explain the difficulties arising in the general case, and our strategy to overcome them. For now, we shall focus on the special case = 4 and further restrict to the discrete group Γ 4,3 .
The key idea in the proof is to exploit the fact that there is a canonical embedding Γ 3,3 ≅ ℍ 5 ( ) ↪ Γ 4,3 of the 5-dimensional Heisenberg group which, as we mentioned before, has Dehn function 2 . We will explain the main steps of the proof and, in particular, where we use the embedding of ℍ 5 ( ): In a first step we reduce to considering null-homotopic words = ( 1 , 2 ) in the generators of the first factor Λ 4 ⩽ Γ 4,3 of the central product. The core of the argument, which we will explain now, consists of transforming ( 1 , 2 ) into a word that closely resembles the normal form 3 3 1 1 2 2 4 4 . Since for a null-homotopic word we must have 3 = 1 = 2 = 4 = 0 we can then conclude from there.
Given a word ( 1 , 2 ) of length ( ) = the idea is to push all 1 's to the left one-by-one, starting with the leftmost one. Modulo 2 (Λ 4 ) this will eventually yield the word 1 1 2 2 . However, whenever we commute a 1 with a ( ) 2 we produce an error term ( ) 3 which we then need to move out of the way. We do this by pushing it to the very left of the word, at the cost of producing a central word of the form [ ( ) 1 , ( ]. All steps up to this point require ( 2 ) relations and repeating this ( ) times, once for each instance of 1 , would provide us with the desired area bound of ( 3 ).
However, the problem is that this is only true modulo 3 (Λ 4 ). Instead we also need to move the word of the form [ ( ) 1 , ( ] which we produced out of the way in every step. We want to do this by moving it to the very right of the word. This involves commuting it with words of the form ( ) 1 , which in the 3-Heisenberg group ℍ 3 ( ) ≅ Λ 3 = ⟨ 1 , 3 ⟩ requires ( 3 ) relations. After ( ) repetitions we would thus end up with an upper area bound of ( 4 ) rather than ( 3 ). This is the point at which we make fundamental use of the fact that the group ⟨ 1 , 3 ⟩ is the left factor of an embedded 5-dimensional Heisenberg group obtained by taking the central product of Λ 3 with itself. Indeed, this allows us to replace the central word [ ( ) 1 , ( ) 3 ] in the left factor by a central word of the same length in the right factor of the central product Γ 3,3 using ( 2 ) relations. We can then commute with ( ) 1 using only ( 2 ) relations. After ( ) repetitions of the total process, each of which has a total cost of ( 2 ) relations, we thus reach a word that closely resembles the normal form 3 3 1 1 2 2 4 4 . For this we required only ( 3 ) relations, rather than the expected ( 4 ) relations, and we can conclude from there. Note that in fact in this last step we use that ℍ 5 ( ) has Dehn function 2 once more to simplify a product of ( ) copies of central words of the form [ ( ) 1 , ( ] into the trivial word. We will use various analogues of both of the kinds of above transformations coming from the embedded copy of ℍ 5 ( ) ⩽ Γ 4,3 for general , by exploiting the embedded subgroup −1, −1 ⩽ , −1 . They will appear at many points of the proof and ultimately lead to two key technical results: the Main commuting Lemma (Lemma 6.2) and the Cancelling Lemma (Lemma 6.9), which in essence can be seen as our most general versions of the first and second application of ℍ 5 ( ) above. There will be various challenges to overcome for general in comparison to = 4. The most obvious one is that the central series of Λ has more than three non-trivial terms. This means that there is not enough space to mimic the trick we used for = 4, where we conveniently left terms in 1 (Λ 4 ) in the middle, moved terms in 2 (Λ 4 ) to the left and finally moved terms in 3 (Λ 4 ) to the right, which provided us with a suitable normal form. When computing the upper bounds for the Dehn functions of the , −1 we will use Dehn functions of compact presentations rather than either geometric methods or Dehn functions of discrete groups. Indeed, while we do use a more geometric approach in our proof of the lower bounds, we were not able to find an obvious geometric model for our groups that allows for the 'easy' computation of upper bounds on Dehn functions. On the other hand, they are too complicated to pursue a discrete combinatorial approach. It is thus really the hybrid approach between the two points of view provided by compact presentations of Lie groups that allows us to prove our results. Indeed, it provides us with the 'geometric' flexibility of writing our words in a relatively simple and thus manageable form on the combinatorial side, while at the same time allowing us to use all of the classical tools and manipulations from discrete combinatorial group theory, thereby not requiring the use of an intricate geometric model. We thus believe that this kind of approach really merits attention, as it might also be instrumental in other problems in this area. We emphasise that this has also been suggested in [14].

DEHN FUNCTIONS, FILLING DIAMETERS AND FILLING PAIRS
In this section, we will introduce basic notions on Dehn functions, filling diameters and filling pairs and collect some important well-known results on them.

Dehn functions of compactly presented groups
Let be a compactly generated locally compact group. For any compact generating set let ( , ) be the kernel of the epimorphism ↠ where denotes the free group over . Recall that is compactly presented with compact presentation  = ⟨ | ⟩ if ( , ) is the normal closure of ⊂ ( , ) such that is bounded with respect to the word metric on . Simply connected Lie groups are known to be compactly presented (see, for instance, [38,Theorem 2.6]). For simply connected nilpotent Lie groups such presentations can theoretically be obtained over an arbitrary compact generating set from the knowledge of a Lie algebra presentation using the Baker-Campbell-Hausdorff series (of which only finitely many terms actually appear). These presentations are however unpractical to work with and in Subsection 5.1 we shall thus provide explicit constructions of compact presentations for the groups and , . Let  = ⟨ | ⟩ be a compact presentation of a locally compact group . Recall that a freely reduced word over represents the identity in if and only if it belongs to the normal closure of . Further recall that we call such a word null-homotopic, that we define Area( ) as the minimal number of conjugates of relations ∈ ±1 whose product is freely equal to and that the Dehn function  of a compact presentation  is defined by  ( ) = sup {Area( ) ∶ null-homotopic and freely reduced of length ⩽ }.
Remark 3.1. Two remarks are in order here. First, it is easy to check that provided that it is finite, the asymptotic behaviour of  ( ) does not depend on a choice of compact presentation. Second, by definition any compactly presented locally compact group admits a presentation of the form ⟨ | ⟩ where = consists of all null homotopic words in of length at most and for any such presentation  is finite [8,Proposition 11.3].
It turns out that the Riemannian definition of the Dehn function that we gave in the introduction and the combinatorial definition have the same asymptotic behaviour. More generally, given a Riemannian manifold define ( ) to be the supremum of areas needed to fill loops of length at most in . The following result is due to Bridson when is discrete [6, Section 5].

Proposition 3.2 [13, Proposition 2.C.1]. Let be a locally compact group with a proper cocompact isometric action on a simply connected Riemannian manifold . Then is compactly presented and the Dehn function of satisfies
To complete the picture we mention that the asymptotic behaviour of the Dehn function is invariant under quasi-isometry; this was proved for finitely presented groups in [2] and the proof adapts without changes to compactly presented groups.

Fillings in balls of controlled radius
We will be interested in constructing fillings where we simultaneously control the number of relations and the diameter of the image of the corresponding van Kampen diagram. Geometrically this amounts to filling a word in a ball of controlled radius. As in the previous section let  = ⟨ | ⟩ be a compact presentation of a locally compact group . We will say that a word = ( ) has (word) diameter ⩽ in if the associated path in the Cayley graph of stays at distance ⩽ from the identity 1 ∈ . Equivalently, has diameter ⩽ if for any decomposition = 1 ⋅ 2 into two subwords we have dist Cay( , ) (1, [ 1 ]) ⩽ . of area has (filling) diameter if has word diameter ⩽ for 1 ⩽ ⩽ .
We will often drop the specification 'word' and 'filling' diameter when it is clear from the context which one we mean.
We will say that two words ( ) and ′ ( ) are equivalent with area (or at cost) and diameter if ′ ⋅ −1 is null-homotopic and admits a filling with area and diameter . In this case we will also say that the identity ≡ ′ holds with area and diameter in .
Remark 3.4. We emphasise that the definition of the diameter of the equivalence ≡ ′ involved a choice: we chose to estimate the diameter of a filling of ′ ⋅ −1 rather than ′−1 ⋅ . While both words have the same filling areas they differ by a conjugation by ′ and thus their filling diameters can differ by ( ′ ). We shall stick to this choice throughout the paper.
We will frequently use the following simple observation: Then the identity ≡ ′ mod ⟨⟨ ⟩⟩ for ′ = 1 ′ 2 3 holds with area and diameter ′ ⩽ + in , where is the word diameter of 1 . In particular, if ⩽ and ⩽ , then ′ ⩽ 2 .
Proof. This follows easily from the definitions. □ We call a word 1 (respectively, 3 ) as in Lemma 3.5 a prefix (respectively, suffix) word for the transformation of into ′ .
Remark 3.6. The fact that only the prefix word 1 plays a role in the estimate in Lemma 3.5 comes from the choice we discussed in Remark 3.4.

3.3
Filling pairs Definition 3.7. Given two increasing unbounded functions , g ∶ + → + , we say that a compactly presented group admits a ( , g)-filling pair if every null-homotopic word = ( ) of length has a filling of area in ( ( )) and filling diameter in (g( )).
Filling pairs are quasi-isometry invariants of compactly presented groups up to equivalence ≍ (where for hyperbolic groups we allow for a linear term in the first entry). The proof is the same as for Dehn functions and we refer to Lemma 9.7 for details, where we prove a more general result for SBEs.
If is a topological group, recall that < is a retract of if it is a closed subgroup and there is a surjective homomorphism ∶ → which restricts to the identity on . The following are well-known in the context of Dehn functions of finitely presented groups (see [3,Lemma 1], respectively, [5, Proposition 2.1]) and their proofs adapt easily to filling pairs of compactly presented groups. Lemma 3.8. Let be a compactly presented locally compact group. If is a retract of , then is compactly presented and any filling pair for is a filling pair for . Lemma 3.9. Let 1 and 2 be noncompact compactly presented locally compact groups. Let = 1 × 2 and let ( 1 , g 1 ) (respectively, ( 2 , g 2 )) be filling pairs for 1 (respectively, 2 ). Then is a filling pair for .

WARM UP -AN UPPER BOUND FOR THE DEHN FUNCTION OF
, As a warm up for the general proof of the upper bound of −1 on the Dehn function of , and , −1 we will discuss the special case when = 4. This case will serve as base case for our induction argument in Section 6. The case of general is very subtle, requiring a careful chain of technical lemmas. In contrast the case = 4 captures much of the essence of how our general proof works, while avoiding almost all of the technical difficulties. In particular, we can work hands on with the finitely presented lattice Γ 4,3 . We will conclude this section by explaining the difficulties we will face when dealing with general values of and how we will resolve them.

Deriving a cubical upper bound for ,
As recalled in the previous section, the Dehn functions of Γ 4,3 and of 4,3 are equivalent, and it will be easier here to deal with Γ 4,3 . Some of the techniques and notation we will use in this section are inspired by Olshanskii and Sapir's combinatorial proof that the Dehn function of the 5dimensional Heisenberg group is quadratic [28]. However, our line of argument is rather different from theirs. Indeed we will start by assuming that 5 ( ) ≍ 2 , which is the main result of their work, and deduce from it that Γ 4,3 ( ) ≍ 3 .
We recall that we work with the presentation We state the following result: Theorem 4.1 [1,28]. Γ 3,3 admits ( 2 , ) as a filling pair.
The linear bound on the diameter is not stated in these references. However, it is easy to deduce it from Allcock's proof. Since he works with the Riemannian version of the Dehn function in the real Heisenberg group, we postpone the presentation of his argument to Subsection 6.8.
The key observation that makes our proof work is that the natural embedding of ℍ 5 ( ) in Γ 4,3 combined with Theorem 4.1 allows us to manipulate words of length in the letters { 1 , 3 , 1 , 3 } at cost ≲ 2 and in a ball of diameter ≲ . The following is a particularly important immediate consequence, as it enables us to 'change between factors' and thus exploit the central product structure of Γ 4,3 . The most important class of central words ( 1 , 3 ) ∈ 3 (Γ 4,3 ) will be words of the form where and are integers and is an integer satisfying 0 ⩽ | | < | |. In a sense they are the discrete prototype for the words Ω that we will introduce in Subsection 5.2 and then use throughout the remainder of the paper. The following observation is straightforward. We record the following simple consequence of Lemmas 4.2 and 4.3: There is a constant 1 > 0 such that for every two words 1 = ( 1 , 1 , 1 ) and 2 = ( 2 , 2 , 2 ), their product 1 ⋅ 2 can be transformed into a word 3 = ( 3 , 3 , 3 ) with From this innocuous observation we deduce the subsequent lemma, which is the second key tool for our proof. We will use it in the case when = , in which it shows that a central nullhomotopic word of the form ∏ =1 has area bounded by 2 3 . In particular, up to constants, its area is bounded by the function ↦ 3 2 in = ( ), rather than by ↦ 2 , as one might a priori expect. Proof. The proof is by induction on , with the result for = 1 being trivial. Assume that the result holds for ⩾ 1 and let ∏ +1 =1 be null-homotopic. Since ≡ + for 1 ⩽ ⩽ + 1 is in the centre of Γ 3,3 it follows that ∑ +1 =1 + = 0. In particular, there is some 0 such that
To obtain an upper bound on the area of ( 1 , 2 ) we will iteratively move all instances of 1 in to the left, starting with the left-most. After moving an 1 to the left we move all 3 's created in the process to the left. As a consequence, we will obtain a word of the form = ( , , 0) with | |, | | ⩽ , which we move to the right.
After the th iteration of this process we may assume that we have a word of the form Since the exponent sum of the 1 's and 2 's is zero, repeating this process ⩽ times will yield a null-homotopic word − is in the centre of Γ 4,3 it follows that it is null-homotopic and thus = 0. We now apply Lemma 4.5 with ∶= to conclude that It remains to explain the + 1th iteration of our procedure and to check that it has quadratically bounded area and linearly bounded diameter. It is here where we will make fundamental use of Lemma 4.2. We will discuss the case +1 1 , the case −1 1 being similar. The following identities hold in Γ 4,3 : (4.8) Setting +1 = ( 2 + 1, − 3 , 0) completes the + 1th step. We remark that in the case −1 1 we obtain new terms + 3 3 and ( 2 , 3 , 0). Using that | 2 | + | 3 | + 1 + ( ( 1 , 2 )) ⩽ we obtain that the number of relations required to obtain consecutive lines of the equation is bounded as follows: In particular, there is a constant 3 > 0 such that the total cost of this transformation is ⩽ 3 2 .
Since we repeat this process ⩽ times, this provides the desired area estimate in Theorem 4.6. [29] (or Lemma 5.13), meaning that the prefix word of all of our transformations has diameter in ( Thus, by combining the linear diameter bounds in Lemma 4.2 and Claim 4.8 with Lemma 3.5, we obtain that all of our transformations satisfy a linear diameter bound, completing the proof of Theorem 4.6.

Developing a strategy for the proof for general
In some sense what made our proof work for = 4 is that this degree is low enough so that we could conveniently shift powers of 3 to the left, central words of the form ( , , ) to the right and keep the remainder of our word in 1 and 2 in the middle. This allowed us to elegantly avoid and hide a key difficulty that makes any brute force attempt to generalise our approach to arbitrary values of fail: the distortion of terms in (Γ , −1 ) being 1 , the cost of 'naively' creating and reordering powers of the will be much too high. On the other hand, the commuting trick exploiting the second factor (generated by the 's) will only work for central words.
We overcome these difficulties through a sequence of results that on the surface seem like a long list of technical lemmas, but really follow a concrete strategy designed to avoid the above obstacles. Moreover, it will turn out to be of great use to switch to the setting of compact presentations and work in the real Malcev completion , −1 rather than in the discrete group Γ , −1 . But for now let us pretend we work in Γ , −1 . For ⩾ 1 and = ( 1 , … , ) ∈ ℤ , we let Ω ( ) be the following word in 1 and 2 We observe that Ω ( ) corresponds to an element of the th term of the lower central series of the free group generated by 1 and 2 . In particular, for = it defines a relation in Λ , and therefore in Γ , and Γ , −1 . The non-technical key steps of our proof for general are as follows.
Step 0: Similar arguments as above allow us to reduce to words ( 1 , 2 ).
Step 1: We use the results on efficient sets of words presented in Subsection 5.3 to argue that we can reduce to null-homotopic words of the form with | |, | | ⩽ and uniformly bounded by some constant > 0.
Step 2: By shifting the 1 's to the left in blocks, we transform the word into a product of ⩽ ′ iterated commutators of the form Ω ( ) ±1 , with 2 ⩽ ⩽ − 1 and ∈ , and order them by the size of the (for a suitable constant ′ > 0). This provides us with a word of length ≲ that (at least morally) is very similar to a word in the Malcev normal form of Subsection 5.1.
Step 3: We consecutively merge all terms of the form Ω ( ) for increasing , starting with = 2. Using that is null-homotopic this process will terminate in the trivial word. At any stage we will make sure that the remaining word stays of length ≲ .
Note that for technical reasons the above steps do not appear in the precisely same order in Section 6. However, keeping them in mind when reading the proof should be helpful in understanding its structure.
The most difficult steps are Steps 2 and 3. Performing them essentially requires us to be able to do two things at sufficiently low cost.
(1) Merge two words of the form Ω ( 1 ) and Ω ( 2 ) into a new word of a similar form and of length ≲ . (2) Commute certain types of words. In particular, we will have to commute words of the form Ω 1 ( 1 ) with words of the form Ω 2 ( 2 ) at cost ≲ −1 . The bulk of the technical work in Section 6 is concerned with resolving these two problems. Concretely, (1) will be resolved by Lemma 6.9, which we will often refer to as the Cancelling Lemma, while (2) will be resolved by Lemma 6.2, which we will often refer to as the Main commuting Lemma. Note that the Cancelling Lemma and the Main commuting Lemma are in some sense beefed-up and considerably harder to prove versions of Lemma 4.5 and of the commutation of terms enabled by Lemma 4.2.
In fact we will first prove the Main commuting Lemma and then the Cancelling Lemma, as the former will be required in the proof of the latter. The proofs of both will be by a rather subtle double induction in and and will be divided into several auxiliary technical lemmas. Throughout the proofs of these results we will rely heavily on applying the fact that, by induction, Γ −1, −1 ( ) ≍ −2 to rewrite words in the generators of the canonically embedded subgroup Γ −1, −1 ↪ Γ , −1 . Similar to the use of Lemma 4.2 in Subsection 4.1, we will also make essential use of the fact that we can replace words of length in 1 and 3 that are contained in −1 (Γ −1, −1 ) by words in 1 and 3 at cost ≲ −2 , to enable us to commute them with words in the at a low cost. In particular, we will use this to start the induction in some of the technical Lemmas leading up to the Main commuting Lemma.

PRELIMINARIES FOR THE GENERAL CASE
In this section, we set the stage for the proof of the upper bound on the Dehn functions of , and , −1 for general . In Subsection 5.1, we start by constructing explicit compact presentations. In Subsection 5.3, we recall the notion of efficient words, which will allow us to restrict to certain families of simpler words when proving upper bounds on the Dehn functions. We then explain how to obtain such a set of efficient words with respect to our presentations. Finally, in Subsection 5.4, we prove some technical results that we will require in Section 6 to compute upper bounds on diameters of fillings.

5.1
Compact presentations of the groups , and , Recall from the introduction that Λ denotes the model filiform group with presentation and denotes its real Malcev completion. The group Γ , is defined as the central product of Λ with Λ for 3 ⩽ ⩽ . We deduce the following finite presentation of Γ , : Observe that for Γ , −1 we purposefully used the notation 1 , − +2 , … , instead of 1 , 2 , … , −1 as it allows us to see Γ , as a subgroup of Γ , . Actually, it will be more convenient to work with compact presentations of their respective Malcev completions , . We describe below a way to deduce a compact presentation of the group from a finite presentation of a lattice.
Let Γ be a finitely generated torsion-free nilpotent group. Then Γ is strongly polycyclic, that is, admits a composition series Γ = 0 Γ ⊳ 1 Γ ⊳ ⋯ ⊳ Γ = {1} with Γ∕ +1 Γ = . It can be chosen to refine the lower central series, that is, there exist integers such that Γ = Γ for all with suitable . Choosing representatives of the generators of the quotients Γ∕ +1 Γ, one can build a generating set = { 1 , … , } such that [ , ] ∈ ⟨ +1 , … , ⟩ whenever < and every ∈ Γ uniquely writes as 1 1 ⋯ with ∈ . is called a Malcev basis for Γ. With respect to the integer coordinates one can prove that the multiplication law is polynomial, that is, that there are polynomials 1 , … , ]. An effective way of constructing the Malcev completion of Γ is to extend this polynomial law (denote it ⋆) from to . Let be any simply connected nilpotent Lie group containing Γ as a lattice. Then the isomorphism Γ → ( , ⋆) extends to an isomorphism → ( , ⋆). This can be established independently of the existence part of the Malcev theorem [35, Corollary 2, p. 34] by Zariski-density arguments.
We shall use the following notation throughout: for ∈ Γ and ∈ we denote = exp( log ) and for all subsets ⊂ and > 0 we define The subsequent result explains how one can obtain a compact presentation for a simply connected nilpotent Lie group starting with a Malcev basis of a lattice Γ < .

Proposition 5.2. Let Γ be a lattice in a simply connected nilpotent Lie group and let
such that for all , ∈ the following equality holds in Γ: Proof.
(1) is a direct consequence of the existence of the polynomials 1 , … , and the construction of b from a refinement of the lower central series. For (2) note that these relations allow us to transform any word over b into its Malcev normal form 1 1 ⋯ . Finally, we prove (3) in three steps.
• b is a generating set: This is clear from the isomorphism → ( , ⋆). Moreover, b is compact as image of a compact set under the exponential map. On the other hand, is a polynomial function by the Baker-Campbell-Hausdorff formula. We deduce that it is identically 0 on 2 and therefore that , ( , ) holds for all , .
• As in (2) the relations in allow us to transform any product of powers of elements in b into its normal form 1 1 ⋯ . Hence, the normal subgroup of generated by coincides with ker( b → ). □ Remark 5.3. Compact presentations offer a technical advantage over finite presentations when manipulating words as they allow to reduce length. For instance, representing a central element in 5 ( ) by a short length word over b needs a product of two commutators due to divisibility issues (compare [28] and Subsection 4.1) while a single one is sufficient over b .
Remark 5.4. For our purposes it will suffice to consider only the case = 1 and we will restrict to it in Section 6. However, producing a presentation for general is no harder and might be useful for future applications. Hence, we write our results in this general context in this section.
Convention. From now on we will omit the relations ( , ) from our compact presentations to simplify notation, as they are rather self-explanatory.
To obtain an explicit compact presentation for , we compute the polynomials , corresponding to the Malcev basis b .
Proposition 5.5. For , ∈ the following relation holds in : In particular, let = { 1 , 2 } and b = { 1 , … , −1 , }. Then for every > 0 the set is a compact generating subset of and the latter admits a compact presentation  ( ) given by the generating subset b and the relators Proof of Proposition 5.5. It suffices to prove the formula and area estimate for = 2 since ⟨ 1 , ⟩ ≅ Λ +2− with 1 ↦ 1 and 2 ↦ defines an isomorphism. The first step is to prove [ 1 , 2 ] = 3 for every ; this is obtained by induction on (for an integer) and we deduce the area and diameter estimates ( 2 ) and ( ), respectively. We now assume the formula for ( , ), denoting its area by Area( , ), and consider +1 1 2 . In the following calculation we record the cost on the right.
We provide some explanations for our transformations: on the third line the rightmost 1 is brought to the left which creates -terms for ⩾ 4; they are gathered with the previous ones. On the fourth line the rightmost 3 is brought to the left and no new term is produced since 3 commutes with all the for ⩾ 4.
We deduce from our estimates that where and ′ are positive constants, and thus that Area( , ) = , ( −1 2 ) by induction on .
For the diameter bound observe that the th term of the lower central series is 1 -distorted [29] (see also Lemma 5.13). Thus, all prefix words of transformations appearing above have diameter in , (| | + | |) and we conclude by Lemma 3.5 that our filling for (5.1) has diameter ≲ , | | + | |. Finally, the remaining properties follow from Example 5.1 and Proposition 5.2. □ Combining Proposition 5.5 and the fact that , is the central product of with , we deduce the following compact presentation of , .

Corollary 5.6. For 3 ⩽ ⩽ , a compact presentation of , is given for every
We end this section by recalling the following well-known free equalities that hold in every group and that we will require at many points throughout the remainder of this work. Lemma 5.7. Let be a group and let , , be words in some generating set for . Then the following free identities hold:

A family of special words
We now introduce a family of words that will play a crucial role in the following sections. For ⩾ ⩾ 2, ⩾ 1 and = ( 1 , … , ) ∈ we let Ω ( ) be the word for ⩾ 2 and 1 if = 1. For = 2, we shall simply denote it † by Ω ( ).
We observe that although Ω ( ) is a priori defined as a word in ∞ we can view it as an element of by identifying with a product of ⌈| |∕ ⌉ letters of the form 1 with | | ⩽ . In what follows such identifications will be made implicitly. Using that , −1 is ( − 1)-nilpotent and Proposition 5.5, we easily deduce the following useful identities.
In particular, for all

Reduction to products of efficient words
We build here on [14]. We let be a finite alphabet and let denote the free group on . Given a subset  ⊂ and an integer ⩾ 1, we denote  [ ] the collection of concatenations of at most words in  .

Definition 5.9. Given an integer
Given a set  of words in , we shall say that we have a filling pair ( , g) for in restriction to words in  if every relation of length that lies in  admits a filling of area in ( ( )) and filling diameter in (g( )).
The following is based on an original observation of Gromov [ Proof. The statement of [14,Proposition 4.3] is that is an isoperimetric function for . However, it is easy to deduce from its proof that ( , ) is a filling pair. Indeed, the proof consists of filling a loop of length using loops of length in ( ∕ ) and a loop ′ in  [ ] of length in ( ). While the argument used in [14] to obtain the desired area bounds applies for very general functions, it is not hard to check that using their methods one can actually produce a filling of area ≲ by iterating this procedure log ( ) times. In particular, this yields the existence of such a filling of of diameter in ( ∑ ⩾1 ∕ ) = ( ). □ We recall that = { 1 , 2 } ⊂ Λ . We define the subset  ⊂ of all powers of elements in : The main result of this section is Proposition 5.11. For all ⩾ 3 and > 0 the subset  is (1)-efficient with respect to the compact presentation  ( ) of provided by Proposition 5.5.
We immediately deduce the following corollary, which is the statement we shall need in our proof of the upper bound of the Dehn function. Define Cyclic subgroups of the th term of the descending central series have distortion in 1∕ ( [29]). The following lemmas provide related estimates that will be required in various places of our proof.
In particular, is a generating subset of .
Proof. The proof is by descending induction on . By Lemma 5.8, we have proving the case = .
Now assume that the result holds for = 0 + 1 and let ∶= 1 0 −1 . Observe that an iterated application of Proposition 5.5, Lemma 5.7(2) and the fact that is metabelian to the innermost commutator yields the following identities in (that is, modulo ): Because ( ) is a polynomial of degree in , we deduce that for any choice of 3 ⩽ 1 < ⋯ < 0 −2 ⩽ the exponent of 0 −2 is a polynomial of degree 0 −2 − 1 in . Since there are only finitely many terms for each index 0 ⩽ 0 −2 ⩽ , we deduce that there are polynomials ( ) of degree − 1 for Finally, an explicit evaluation shows that The result now follows by applying the induction hypothesis to the − ( ) . □ Lemma 5.14. For 1 , … , , 1 , … , ∈ let = 1 2 1 1 ⋅ ⋯ ⋅ 2 1 and let ∶= Proof. We will move all 1 's in to the left to put the word in normal form. Setting 0 = 0 and introducing the notation e ∶= ∑ = we first observe that, by Proposition 5.5, the identity holds in for 1 ⩽ ⩽ . Thus, moving powers of 1 to the left, starting with the rightmost one, and [ , ] = 1 for , ⩾ 2 imply that mod ⟨⟨ ⟩⟩.
. Using that ( ) is a polynomial of degree in and that |e | ⩽ , it is now easy to deduce that | | = ( −1 ) + (1). This completes the proof. □ We will now explain how to derive Proposition 5.11 from Lemmas 5.13 and 5.14.

Upper bounds on diameters
We conclude this section by recording a few results which we will require to show that all fillings in Section 6 have linearly bounded diameter.
Proof. The subgroup of generated by 1 and is isomorphic to − +2 and there is a canonical embedding − +2 ↪ induced by an embedding of presentations. Thus, by Lemma 5.14 for , . In particular, By Lemma 5.13 there is a word = +1 ⋅ ⋯ ⋅ with This completes the proof. □ Corollary 5.16. Assume that in Lemma 5.15, is a positive integer such that ⩽ for 1 ⩽ ⩽ . Then in the conclusion we obtain .
In particular, for ⩽ and = 3 we deduce that |g| ≲ .
and has word diameter ≲ .
In a second application of Lemma 5.15 we will require the following estimate. such that the following inequality holds: First observe that by definition of : Since − + 1 ⩽ 0 and ⩾ we now deduce from the geometric series that This completes the proof. □ In this section, we will derive upper bounds on the Dehn functions of , and , −1 . In Subsection 6.1, we state a sequence of auxiliary results and explain how they are used to prove the desired upper bounds by induction on . This will be visualised by Figure 1. In the remaining sections, we then prove these results in the described order, finishing with the proof of the main result in Subsection 6.8.

Main theorem and structure of the proof
The goal of this section is to prove the following key result of our paper. The proof proceeds by induction on . We will see that for both groups we can reduce to nullhomotopic words of the form ( 1 , 2 ), where 1 and 2 generate the first factor (see Subsection 6.8). In view of the canonical embedding , −1 ↪ , , we deduce that it is enough to show that , −1 ( ) ≼ −1 (see Lemma 6.31). The core of the proof consists in deducing from −1, −1 ( ) ≼ −2 that , −1 ( ) ≼ −1 . We recall the following notation, for every > 0: and  ∶= { | ∈ , ∈ } and  ∶= { | ∈ , ∈ }.
We will fix = 1 once and for all and will omit the prefix in all expressions, as one fixed choice for will suffice for the remainder of our proof (cf. Remark 5.4).
By Propositions 5.10 and Corollary 5.12, it suffices to prove that for every > 1 we have is the set consisting of all words obtained by concatenating at most words from the set  (respectively, from  ).
We now describe the structure of the proof via a list of technical lemmas. In what follows, saying that an identity between words in holds in , −1 will be shorthand for saying that it holds in ( , −1 ). It is easy to deduce from its presentation that , −1 is a metabelian group. The first important step is to prove that the commutation relations in , −1 , induced by its metabelian structure, have area ≲ −1 and diameter ≲ . More generally we prove This result will be the consequence of four more specific lemmas. Before stating them we shall recall and introduce some additional notation.
We record the following key observation.
Proof. This is a direct consequence of the fact that there is a canonical embedding of presentations , and , −1 are bounded by −1 rather than . Indeed, it allows us to 'push' words in the first factor which represent central elements into the second factor at a cost that is bounded by the Dehn function of −1, −1 . Using that the commute with the we can then commute them with words in the at a lower cost than one might a priori expect. We use Lemma 6.3 at various points and, in particular, in the proof of Lemma 6.5 to kick-start our induction step from − 1 to . As mentioned above, the Main commuting Lemma 6.2 will result from four sublemmas, dealing with specific commuting relations involving words of type Ω . These lemmas depend on a parameter ⩽ − 1. By -lemma, we mean the statement of the corresponding lemma for a specific value of .
The first one deals with commutators of words of type Ω with words representing elements of the derived subgroup. Our second lemma treats commutators of words of type Ω with powers of 2 .
The following lemmas are versions of Lemmas 6.4 and 6.5 for Ω instead of Ω . To prove the Main commuting Lemma 6.2, we shall need a further reduction step, reducing to words of bounded length in elements of type Ω .
The Main Theorem 6.1 will be a consequence of the Reduction Lemma 6.8 and the following more subtle technical result, which deals with products of Ω -terms with different values of .
Finally, we record the following technical result which plays a key role at various stages of the proof. The way the Cutting in half -Lemma is used throughout the proof is a bit subtle: we will require its version for − 1 as part of the induction step when proving the commuting -Lemmas for . On the other hand, its -version will be obtained as a corollary of the Main commuting Lemma 6.2 that results from the four commuting -lemmas. Finally, its -version will be instrumental in the proof of the Cancelling -Lemma 6.9 for .
In the proof of the Cutting in half -Lemma we will use the following immediate consequence of the Main commuting Lemma 6.2 for ⩾ 5 (respectively, Theorem 4.6 for = 4). We record it here, as we will require its ( − 1)-version in the proof of the Third and Fourth commuting -Lemmas for ⩾ 5.  has area ≲ −1 and diameter ≲ . For = 4 the same area and diameter estimates follow from Theorem 4.6.
Regarding the proof of the diameter bounds we will adopt the following.
Convention. Throughout this section, the diameter bounds for our fillings will follow from Lemma 3.5. In most cases this will be obvious, since the transformations used, as well as their prefix words, will satisfy evident linear diameter bounds. To keep the proofs as simple as possible we will only add detailed explanations for the diameter bounds where this is not the case.
Initial step of the induction. As explained in Figure 1, the initial step (for = 4) only needs to be settled for the Main Theorem 6.1, and the Cutting in half Lemma 6.10. The former is provided by Theorem 4.6 in the case of 4,3 . We also observe that the area and diameter estimates of the Cutting in half Lemma 6.10 follow from Theorem 4.6. Hence, in order to initiate the induction, two facts need to be established.
We now turn to the proof of the Main Theorem 6.1 for 4,4 . It is a direct consequence of Main Theorem 6.1 for 4,3 and Lemma 6.31. However, some explanation is required as the proof of Lemma 6.31 itself relies on two statements: Lemma 6.3 and Corollary 6.24. Lemma 6.3 has a short self-contained proof which has already been given. Corollary 6.24 asserts that the second identity of Lemma 5.8 holds in 4,3 with area ≲ | | 3 and diameter ≲ | |, which is a consequences of the Main Theorem 6.1 for 4,3 .
Induction hypothesis: Throughout the remainder of this section, we will now assume that ⩾ 5 and that the Main Theorem 6.1 and the Cutting in half Lemma 6.10 hold for − 1. In particular, by induction −1, −1 ( ) ≍ −2 and every null-homotopic word of length ⩽ in ( −1, −1 ) admits a filling of area ≲ −1 −2 and diameter ≲ −1 . The way the induction procedure works is explained in Figure 1.

Preliminary results
We will now record a few simple preliminary results which we will require at different points in the subsequent sections.
Proof. Identities (1) and (3) The second identity follows from the first one by replacing by + 1 and rearranging the terms. □ We will also require the following: Proof. This is a direct consequence of Lemma 5.13 and its proof. □ As a consequence of Lemma 6.13 and the induction hypothesis for − 1 we obtain: Since the right-hand side of (6.4) is trivial we must have ∑ =1 = 0. In particular, there is Since this is an identity of length ≲ in −1, −1 it has area ≲ −2 and diameter ≲ . We can thus reduce to a null-homotopic product of − 1 terms such that every factor is of length ≲ and equal to with | | ≲ −2 . Repeating this argument a further − 1 times shows that our initial word can be reduced to the trivial word at cost ≲ ⋅ −2 . Noting that by Corollary 5.16 all prefix words of our transformations satisfy the asserted diameter bound of + ∑ =3 ( ⋅ −2 ) 1 −1 completes the proof. □ We finish with two more technical results which we will require later. with words ( 1 , 2 ) in the derived subgroup of , −1 using the First commuting -Lemma 6.4 (see Subsection 6.3). We end this section with the following converse of Lemma 6.13.
Proof. This is an immediate consequence of Lemma 5.14. □

First and Second commuting -Lemmas
For simplicity of notation, we will assume that = 3. The proof for > 3 is the same. Recall that we are proceeding by induction on as shown in Figure 1, that is, the Main commuting Lemma 6.2 (which is a special case of the Main Theorem 6.1) and the Cutting in half Lemma 6.10 can be used in the group −1, −1 with area ≲ −2 and diameter ≲ . Note that formally the Cutting in half Lemma 6.10 is stated in the subgroup −1, −2 ⩽ −1, −1 . However, the natural inclusion of the corresponding presentations means that it also holds in −1, −1 . A crucial step in the proof of the First commuting Lemma 6.4 will be the following technical result, allowing us to cut Ω 3 ( ) into pieces. We note that Ω 3 ( ) is a word in 1 and 3 which therefore belongs to −1, −1 . and + 1 ⩽ ⩽ − 2. In , −1 , the area of this identity is ≲ | | −2 and its diameter is ≲ | |. Moreover, the word diameter of is ≲ | |.
Proof. Note that for = 1 this is obvious and the error terms are trivial. The proof for ≠ 1 is in ⌈log 2 ( )⌉ stages (see Figure 2). At the th stage we will be left with 2 terms of the form Ω 3 ( ∕2 ) interlaced with 2 ( −1) error terms , for 1 ⩽ ⩽ . By Lemmas 5.15 and 5.17 the diameter of this word is for a constant (1) as in Lemma 6.10. The same reasoning shows that the word diameter of the word obtained at every stage is ≲ | |.
We apply the Cutting in half Lemma 6.10 for − 1 to each of the words Ω 3 ( ∕2 ) starting with the right-most one; it holds by induction hypothesis. As a consequence we obtain 2 ( +1) words of the form Ω 3 ( ∕2 +1 ) and 2 error terms of the form , +1 . By the Cutting in half Lemma 6.10, F I G U R E 2 Sketch of the steps leading to the fractal form with ∶= ⌈log 2 (| |)⌉ in Lemma 6.17. It omits the error terms , for simplicity. Lemma 3.5 and (6.5), the total area and diameter of the identities performed in the ( + 1)th iteration are ≲ | 2 | −2 and ≲ | |, respectively. After ⌈log 2 (| |)⌉ iterations we obtain the asserted word of word diameter ≲ | |.
The total area of all identities used in the proof is where the last inequality follows since the sum is a convergent geometric series. Indeed, by assumption, ⩽ − 3 and thus − ( − 2) < 0. This completes the proof. □ Proof of the Second commuting k-Lemma 6.5. Observe that the Second commuting ( − 2)-Lemma is an easy consequence of Lemma 6.3 and the fact that [ , ] = 1 ∀ , . We now assume that for ⩽ − 3 we proved the Second commuting ( + 1)-Lemma by induction. We estimate the area and diameter of the null-homotopic word [Ω 3 ( ), 2 ]. By the Fractal form Lemma 6.17 we have where is a word that is a product of ≲ | | terms of the form Ω 3 ( 2 ⌈log 2 (| |)⌉ ) and, for 1 ⩽ ⩽ ⌈log 2 (| |)⌉, 2 ( −1) error terms , ; the terms are in no specific order and we will thus commute them with 2 one-by-one.
Note that, by the Fractal form Lemma 6.17, identity (6.6) has area ≲ | | −2 and diameter ≲ | |. Moreover, has word diameter ≲ | | and thus the same holds for any of its prefix words. Since all transformations used in the remainder of the proof will consist of commuting a piece of the word with 2 and will have diameter ≲ | | + | |, the diameter bound of ≲ | | + | | in the Second commuting -Lemma will follow from Lemma 3.5.
Observe that the word Ω 3 ( 2 ⌈log 2 (| |)⌉ ) has length in (1) so that the area of [Ω 3 ( 2 ⌈log 2 (| |)⌉ ), 2 ] for | | ⩽ 1 is in (1). Thus, the total cost of commuting the | | terms of the form Ω 3 ( where for the last inequality we use that ⩽ − 3 by assumption. We now estimate the cost of commuting the error terms , with 2 . For this we distinguish the cases = − 3 and < − 3, starting with the former. An error term −3, consists of (1) To move it past 2 we use the second factor of our central product: by Lemma 6.3 the identity Considering that there are 2 ( −1)( −3) copies of −3, in we thus obtain the following upper bound for the total cost of commuting all of the error terms with 2 : where to obtain the first inequality in the last line we observe that This completes this step of the proof for = − 3.
To complete the same step of the proof for < − 3 we now assume that by induction the Second commuting -Lemma holds for − 3 ⩾ ⩾ + 1. In this case an error term , is equal to a product of (1) words of the form Ω 3 ( 2 ) ±1 with | | ≲ | | and + 1 ⩽ ⩽ − 2, and there are 2 ( −1) error terms of the form , .
By the Second commuting -Lemma for ⩾ + 1 the total cost of commuting the , with 2 is thus bounded by ) .
Summing up the total cost for all steps in this proof we obtain that This completes the proof of the Second commuting -Lemma. □ After estimating the cost of commuting Ω 3 ( ) with 2 we now need to estimate the cost of commuting Ω 3 ( ) with a word in  [ ].

Third and Fourth commuting -Lemmas
Both lemmas will be easy consequences of the first two commuting -lemmas and the following result: Proof. This is a direct consequence of the 'in particular' part of Corollary 5. 16. □ The key step in the proof of Proposition 6.21 is summarised by the next result.  ] holds in , −1 with area ≲ −2 and diameter ≲ . This completes the proof. □ Proof of Proposition 6.21. We will assume that −1 ⩾ 0, the proof for −1 < 0 being similar. The proof is by induction on ⌊ −1 ⌋. The case ⌊| −1 |⌋ = 0 is trivial, so assume that ⌊ −1 ⌋ > 0. By Lemma 6.12(2), the identity holds in , −1 at cost ≲ | | 2 and with diameter ≲ | |. Applying Lemma 6.23 and the ( − 1)-version of Remark 6.11 a total of − 2 ⩽ − 3 times to Ω ( ) we obtain the identities Ω ( ) ≡  We also record the following useful consequence of the arguments presented in this section. Proof. The identity itself follows from Lemma 5.8. On the other hand, for = − 1, Proposition 6.21, Proposition 5.5 and Lemma 6.15 yield ′ , ′′ ∈ −2 with | ′ |, | ′′ | ≲ | | such that the identity holds in , −1 with area ≲ | | −1 and diameter ≲ | |. Combining these two identities for Ω −1 ( ), we deduce the area and diameter estimates from Lemma 6.14. Proof. Assertion (2) is an immediate consequence of the Fourth commuting -Lemma 6.7. We turn to the proof of (1). We focus on the case −1 ⩾ 0, the case −1 < 0 being similar. By Proposition 6.21 and Addendum 6.22 the identities hold at cost ≲ | | −1 and with diameter ≲ | |. In fact Addendum 6.22 shows that the last word has word diameter ≲ | |. We will implicitly use this in all further diameter estimates of this proof. Using that Ω 3 −1 ( 2 , … , −2 , + , ) ≡ 1 in −1, −1 at cost ≲ | | −2 and with diameter ≲ | |, and applying Lemma 6.12(3), we obtain that From Lemma 6.15 and the fact that Ω −2 (⋅) ≡ 1 in −1, −1 for ⩾ 4 we can now deduce that with cost ≲ | | −1 and diameter ≲ | | (where we use the identification = to simplify notation in the first line).
Applying the induction hypothesis first to (Ω +1 ( 1 , )) ∓1 (if ⩽ − 2) and then to Ω ( ) ±1 yields an identity of the form where the last identity holds with area ≲ −1 and diameter ≲ by Lemma 6.25 (2). We apply Lemma 6.26 and obtain that 1 + 2 1 1 + 2 2 Ω 2 ( 2 , 1 ) −1 3 1 ⋅ ⋯ ⋅ 1 2 ≡ 1 + 2 1 1 + 2 2 Proof of the Main commuting Lemma 6.2. The case where both 1 and 2 are powers of 2 is obvious. Else, we may assume that 1 is in the derived subgroup. We then apply Lemma 6.8 to rewrite it as a product of , (1) many terms of type Ω ±1 , with ⩾ 2, and we conclude thanks to the Third commuting -Lemma 6.6 if 2 is in the derived subgroup and the Fourth commuting -Lemma 6.7 if 2 is a power of 2 . □

Cancelling -Lemma
The proof of the Cancelling -Lemma is by descending induction on . The Cancelling ( − 1)-Lemma is a straightforward consequence of Lemma 6.14, Corollary 6.24 and Lemma 3.5. Thus, assume that the Cancelling -Lemma holds for all − 1 ⩾ ⩾ + 1. The induction step in the proof of the Cancelling -Lemma is one of the most subtle parts of our proof of Main Theorem 6.1. Our goal is to manoeuvre ourselves into a position where we can use that for a word being null-homotopic implies that in its Malcev normal form in , −1 the exponent sum of the must vanish. In particular, this requires extracting the from the word. Pursuing a naive approach using the Fractal form Lemma 6.17 will lead to a word that consists of powers of that cumulatively have word length −1 , as well as many 'error terms' in the form of short iterated commutators that cumulatively have non-linearly bounded word length. Commuting them using our Main commuting Lemma to assemble the on the left and the error terms on the right would be much too expensive. To circumvent this problem we perform the extraction of powers of using a more intricate procedure which can be seen as beefed-up version of the Fractal form Lemma: rather than producing a word in fractal form we merge error terms whenever we create them and thereby keep their numbers low. We emphasise that it is only at this point of the proof that we can do this, as it will require the -versions of the Cutting in half Lemma 6.10 and the Main commuting Lemma 6.2.
We will now perform the core part of the proof of the induction step from + 1 to , where we overcome the aforementioned difficulties. This will provide us with the following technical result.
Proof. The proof is by an inductive procedure in ∶= ⌈log 2 (| |)⌉. When = 1 the result is an immediate consequence of Lemma 6.13 and our choice of relations, since for = 1 we have | | ⩽ 1. The inductive step is encoded in the following claim. also holds for the element ∈ satisfying ⌈log 2 (| |)⌉ = + 1 with cost at most +1 ⩽ 2 + 2 ( −1) , and diameter at most Before proving the claim, let us see why it implies Lemma 6.27. We immediately deduce that ⩽ 2 1 = (| | ) and Using that < − 1 we deduce that , and therefore that = (2 ( −1) ) = (| | −1 ). So, Lemma 6.27 follows. □ Proof of Claim 6.28. Let ∈ with ⌈log 2 (| |)⌉ = + 1. By Lemma 6.10 for , the identity holds with area ≲ | | −1 and diameter ≲ | |, where ( ∕2) is a product of (1) iterated commutators of the form Ω ( ) ±1 with | | ≲ | | 2 and ⩾ + 1. We apply the induction hypothesis for to each of the Ω ( ∕2) successively, starting with the left-most one and moving error terms to the right. After the th application we obtain an identity of the form Since | | ≲ | | , Lemma 5.13 implies that ⋅ +1 has word diameter ≲ | |. An ( + 1)th application of the induction hypothesis for yields with area and diameter . By applying the Main commuting Lemma 6.2 ⩽ ⋅ 2 times we can commute the terms making up , ( ∕2) with the (Ω ( ∕2)) and obtain the identity in , −1 with area ≲ | | −1 and diameter at most + (| |) (for the latter we use Lemma 3.5 and the fact that | ( +1) +1 | , −1 ≲ | | by the induction hypothesis for and Lemma 5.13). Putting all of the above steps together, we deduce that the identity and diameter at most + (| |).
□ As a consequence of Lemma 6.27 we can complete the proof of the Cancelling -Lemma.
We reduce to the Cancelling ( + 1)-Lemma by applying Lemma 6.27 iteratively to the terms Ω ( , ), 1 ⩽ ⩽ , starting with the left-most one and then moving the error terms right. At the beginning of the 0 th step of this process we will have an identity of the form In particular, Lemma 5.13 implies that all prefix words of transformations will have diameter ≲ . We apply Lemma 6.27 to obtain Recall that the , ( , ) are products of ⩽ terms of the form Ω ( ) ±1 with | | ≲ | , | ≲ and ⩾ + 1. We can thus apply the Main commuting Lemma 6.2 a total of ≲ times to obtain We obtain the identity in , −1 from the original null-homotopic word ( 1 , 2 ) with total area ≲ −1 and diameter ≲ ⋅ . By definition of the , ( , ), after applying the Main commuting Lemma 6.2 at most ⋅ ⋅ ( − ) more times, we obtain with area ≲ −1 and diameter ≲ ⋅ , for suitable , , where e ≲ + for ⩾ + 1.
However, it now follows from the assumption that is null-homotopic, that ∑ =1 = 0. Thus, we have reduced to the cancelling ( + 1)-Lemma for some e ≲ and, by induction hypothesis, the null-homotopic word admits a filling of area ≲ , −1 and diameter ≲ , . Thus, all words satisfying the hypothesis of the Cancelling -Lemma have area ≲ , −1 and diameter ≲ . This completes the proof. □

Proof of the main theorem
We are now ready to complete the proof of the Main Theorem 6.1 for , −1 and , . We start by treating the case = 3, observing that 3,3 = ℍ 5 ( ). The fact that ℍ 5 ( ) has quadratic Dehn function was originally proved by Allcock using symplectic geometry. His proof is short and elegant and actually proves a stronger statement: any smooth horizontal -Lipschitz map from 1 to ℍ 5 ( ) extends to a ( )-Lipschitz map defined on the disk. Here, 'horizontal' has the following meaning: we consider a 'horizontal' distribution defined as orthogonal vector complement of the (1-dimensional) derived subalgebra of 5 ( ), and a path is horizontal if it is tangent to at every point.
Allcock's proof can easily be adapted to show that any -Lipschitz piecewise smooth and horizontal map defined on 1 extends to an ( )-Lipschitz map on the disk. In particular, this applies to 'relation loops', that is, loops that are obtained by concatenation of paths of the form ( ) = (0) , where is an element of the generating set Remark 6.30. This also provides a proof of Theorem 4.1: indeed, ℍ 5 ( ) being a uniform lattice in ℍ 5 ( ), the two groups are quasi-isometric, so we can deduce, for instance, from Lemma 9.7 (with = 0 and = 1) that ℍ 5 ( ) admits a ( 2 , ) -filling pair.
We may thus now complete the induction step for ⩾ 4. In particular, we may assume that −1, −1 admits ( −2 , ) as a filling pair. In the previous sections, we have proved that under this assumption all auxiliary results in Subsection 6.1 hold for and it remains to put them together. Indeed, as we shall now see, the Main Theorem 6.1 for is a straightforward consequence of the -versions of the Reduction Lemma 6.8 and the Cancelling 2-Lemma 6.9.
Observe that, on enlarging (twice) if necessary, we may assume that Ω −1 ( 1 , … , −3 , 1, −2 ) ∈  [ ] and thus that It follows that we may assume that = ( 1 , 2 ) is a null-homotopic word in  [ ], at cost ≲ , −2 and diameter ≲ , . We apply the Reduction Lemma 6.8 to obtain an identity of the form in , −1 with area ≲ , −1 and diameter ≲ , , and = , (1). Since is null-homotopic, the same holds for the right-hand side. By applying the Main commuting Lemma 6.2 at most 2 times we obtain that in , −1 with area ≲ , −1 and diameter ≲ , , where ⩽ for as above. The right-hand side of (6.11) remains null-homotopic. The Cancelling 2-Lemma 6.9 thus implies that the right-hand side of (6.11) has area ≲ , −1 and diameter ≲ , in , −1 . Summing up the total area of all transformations we deduce that is null-homotopic with area ≲ , −1 and diameter ≲ , in , −1 . In particular, we have proved that every null-homotopic word in [ ] of length ⩽ admits a filling of area ≲ , −1 and filling diameter ≲ , . By Proposition 5.10, this implies that , −1 admits ( −1 , ) as a filling pair. This completes the proof. □ The Main Theorem 6.1 for , is a direct consequence of the Main Theorem for , −1 and the following result. Lemma 6.31. Let ( 1 , 2 , 1 , 2 ) be a null-homotopic word in ( , ) with ( ) ⩽ .

SECOND COHOMOLOGY AND CENTRALISED DEHN FUNCTIONS
The centralised Dehn function of a discrete torsion-free nilpotent group can be computed by computing the maximal distortion of a central extension. In Subsections 7.1-7.3, we will explain how this characterisation of the centralised Dehn function can be rephrased algebraically in terms of the existence of a second real cohomology class with certain properties. We then apply this algebraic characterisation in Subsection 7.4 to prove Theorem D and, more generally, to analyse the existence of central extensions of central products of nilpotent groups.
Being an -central extension only depends on the equivalence class of the extension and we now explain how it can be read off from 2 ( , ).
Let be a real nilpotent Lie algebra with Lie group . Recall that to any ∈ 2 ( , ) one associates a central extension of defined over the vector space × by  Proof. We start by proving the equivalences between these statements. Note that in (i') and (ii') we specify that the central extension is a connected Lie group, as 'wild' extensions that do not correspond to extensions of the Lie algebra could potentially exist.
The equivalences between (i) and (ii), respectively, (i') and (ii'), are due to Osin's computation of the distortion of subgroups of nilpotent groups [29]. The equivalence between (ii) and (ii') follows from Malcev's correspondence. The equivalence between (ii') and (ii'') follows from the correspondence between a simply connected nilpotent Lie group and its Lie algebra. Finally, the equivalence with (iii) follows from the correspondence between central extensions and the second real cohomology group.
For the remaining part of the statement first observe that the equivalence in the case when < 2 or < 2 is easy to check. Indeed, this can only happen if all central extensions are by taking direct products. Hence, we may assume that ⩾ 2 (or conversely that there is an -central extension with ⩾ 2).
Given a finite presentation ⟨ | ⟩ of Γ let e Γ = ∕[ , ℜ], where ℜ is the normal subgroup spanned by . Consider the central extension It follows that e Γ is a finitely generated nilpotent group, and is generated by the finite subset (modulo [ , ℜ]). Let ∈ and = cent Γ ( ). This means that there exists an element g ∈ e Γ whose word length with respect to is and such that is the minimal integer such that g can be written as a word of length in the generating set of . In other words, cent Γ ( ) is the distortion of in e Γ. It is a classical fact that the central extension e Γ of Γ is universal in the sense that for any other central extension Γ there exists a morphism e Γ → Γ that extends to a morphism of extensions and induces a surjection between the derived subgroups (see, for instance, [42, Lemma 5] for more details). Hence, is indeed characterised by one of the equivalent statements of the proposition. □

Carnot gradings
We recall from the introduction that (i) a nilpotent Lie group is said to be Carnot gradable if its Lie algebra admits a Lie algebra grading = ⨁ =1 such that Liespan( 1 ) = , and that (ii) to any simply connected nilpotent Lie group we can associate a Carnot-graded Lie group ( ) with Carnot graded Lie algebra with brackets induced by those on . In particular, 1 ( , ) and 1 ( ( ), ) = ( ∕[ , ]) ⋆ are naturally isomorphic. is isomorphic to ( ) if and only if is Carnot gradable, in which case the isomorphism is given by the graded linear isomorphism Φ ( , 1 ) ∶ ⨁ → ⨁ ∕ +1 for any Carnot grading ( ) on the Lie algebra.
We refer to [10, 3.2] for more on Carnot gradings. To simplify notation we use the same letters for the elements of the Lie algebra and the Lie group, even though they do not correspond under the exponential map. We emphasise that in this section we will deviate from the remainder of the paper where we denote the generators of the second factor by − +2 , … , −1 , . This difference in notation is because it proves computationally convenient in the respective parts of the paper. We observe that with respect to our generators Identifying , ∕ +1 , with 1 ⊕ 2 ⊕ 1 ⊕ 2 , for = 1, +1 ⊕ +1 , for 2 ⩽ ⩽ − 2, +1 , for − 1 ⩽ ⩽ − 2, and , for = − 1, we can define the brackets of , and of ( , ) on the same vector space. If = , then , is Carnot-graded, otherwise all the brackets are the same in , and ( , ) except that [ 1 , −1 ] = in , while [ 1 , −1 ] = 0 in ( , ). We deduce that with 1, ⩽ 2, and 1, + 2, = , Proof. Assume (Δ 1 ), (Δ 2 ) and (Δ 3 ) and let ∶ e → be the central extension associated to ; decompose eas a product × . In accordance with the definition of -central extension we must prove that (0, 1) ∈ e. By (Δ 2 ) and (Δ 3 ) we may represent this element as where ( e ) = , respectively, ( e ) = . Note that by (Δ 1 ) we may assume that e ∈ 1, e and e ∈ 2, e, and, since 1, + 2, = for all , we deduce that (0, 1) = To see this note that the elements and then generate an abelian Lie subalgebra of , and [34, Th 3.1] requires that the map * ∶ 2 ( , ) → 2 ( , ) associated to ∶ → be nonzero, which amounts to asking for the existence of a cocycle satisfying (Δ 3 ). We note that Pittet's exponent (Γ) nevertheless coincides with the growth exponent of the centralised Dehn function up to dimension 6 included (see Section 10).
Remark 7.7. In the special case when is greater than the nilpotency class of (that is, when = + 1) the condition (Δ 2 ) is automatic given the assumptions on and (as 1 + 2 > ) and ( + 1)-central extensions are the central extensions of step + 1. For this reason ruling out the existence of -central extensions is a simpler task when = + 1.
When is Carnot gradable we can go further into the description of cohomology classes yielding -central extensions. Let ( ) be the Carnot grading on with representing ∕ +1 . Correspondingly, ⋀ 1 ⋆ = ⋆ can be graded in the following way: for ⩾ 1 we set ( ⋀ 1 ⋆ ) = * Hom( , ) where is the projection to .
The exterior square ⋀ 2 ⋆ is then graded by Since ( ) is a Lie algebra grading on , the differential ∶ ⋀ ⋆ → ⋀ +1 ⋆ has degree 0 with respect to these gradings. In particular, the cohomology group 2 ( , ) is also graded and the cohomology classes of weight under this grading produce -central extensions. We will compute the grading on 2 ( , ) below (see Remark 7.15). However, the groups , that we are considering are not Carnot gradable for ≠ , , ⩾ 3. Thus, our main tool in this section will be the criterion provided by Proposition 7.5.

Central extensions of central products
We refer to the introduction for the definition of a central product × of Lie algebras and (respectively, × of groups and ). Here we will be interested in understanding the existence of central extensions of central products in general and, more specifically, in the context of the central products , . We start with two general results. Note that [e , e] ∈ ⟨ ′ ⟩ for all if (e) ∈ , and that [e , e] ∈ ⟨ ′ ⟩ for all if (e) ∈ . Since ′ is central it follows that, for ⩾ 3, -fold commutators of e and e vanish, unless they only contain e 's (respectively, e 's). Indeed, the only commutators where this is not trivially true are the [e 1 , e 2 , … , e ] (respectively, [e 1 , e 2 , … , e ]) and they vanish by the Jacobi identity.

Lemma 7.12 (Betti numbers). Let > ⩾ 3. Then
As before, we will perform our Betti number computations using Lie algebra cohomology. To deduce Lemma 7.12 we will thus invoke the following result, that is due to Matsushima for = 1, 2 and Nomizu for all [35,Corollary 7.28]. It shows that the real cohomology of finitely generated torsion-free nilpotent groups only depends on the real Malcev completion, an early manifestation of Shalom's theorem. Before proving Proposition 7.11 and Lemma 7.12, we observe that they allow us to complete the proof of Theorem D, modulo the lower bound from Section 8.
Proof of Theorem D. The first part is a direct consequence of Lemmas 7.9 and 7.10 and Propositions 7.11 and 7.2. The second part follows from Theorem A, whose proof will be completed in Section 8. □ Proof of Proposition 7.11 and Lemma 7.12. Note that if is a one-form on , then is the two-form such that ( , ) = − ([ , ]) for every , ∈ . We will use this below without further mention when computing differentials.
Let We first prove (7.5). For this we need to compute 2 ( , ). Since this computation is wellknown (it is originally due to Vergne [39]), we only sketch it here and leave the details as an exercise to the reader. We emphasise that this is an exercise well-worth doing to get acquainted with Lie algebra cohomology computations.
While we do not use it at this point we record the following observation; it is well-known to experts.
This observation is interesting in itself and also in view of Section 10. However, most importantly comparing it to (7.21) provides some intuition for why , admits no ( − 1)-central extension when is odd. Indeed, we will see that for odd the analogous cohomology class [ 2 ′ ] vanishes in 2 ( , , ), while it survives when is even. In fact, it is precisely the form that induces the ( − 1)-central extension of , when is even. This is also mirrored by the distinct Betti number discrepancies in Lemma 7.12. Computationally, this difference is reflected in the fact that in we have = − 1 ∧ −1 , while in , we have = − 1 ∧ −1 − 1 ∧ −1 . This ultimately implies that the coefficient 2, of 2 ∧ must be zero for every cocycle in , with odd, while it can be non-zero for cocycles in or in , when is even.
Hence, we may assume that = and therefore ,1 = 1 and ,2 = − 2 for all . In particular, = −1, −1 + and (Δ 2 ) implies that On the other hand, evaluating the sum of the ( , ) yields where for the last identity we observe that the only pair of basis vectors of the form ( * , ) and ( * , −1 ) on which our basis of representatives of cohomology classes does not vanish is ( 1 , −1 ). Comparing the two equalities gives a contradiction. This completes the proof of Proposition 7.11. □ Remark 7.16. The cocyle 2 of the preceding proof was introduced by Vergne in her computation of 2 ( , ) [39]. When ⩾ 5 is odd the central extension associated to the cocycle +1 on F I G U R E 4 Carnot graded filiform Lie algebras ( 3 is the Lie algebra of the Heisenberg group, 6 and ′ 6 are L 6,18 and L 6,16 in de Graaf's list [15]). We use the same notation for the cocycles as in the proof of Proposition 7.11. produces a filiform, but not model filiform, Carnot graded Lie algebra of dimension . Vergne proved its existence and uniqueness (see also Figure 4).
The lower bound of −1 on the Dehn functions of , −1 for even ⩾ 4 (respectively, of , for all ⩾ 4) that we obtain from central extensions is sharp by Theorem 6.1. In contrast, and maybe at first rather unexpectedly, for odd the lower bound of −2 on the Dehn function of , −1 obtained from central extensions is not sharp. In fact not even its exponent is sharp, providing the first example of a group with this property. We will prove this in the next section. There is a moral reason for this discrepancy, which we will exploit in the next section; for an explanation of this we refer to Section 2.

LOWER BOUNDS ON THE DEHN FUNCTION FROM INTEGRATION OF FORMS
In this section, we will explain how to obtain lower bounds on the Dehn functions of the , by integrating bounded forms on Lie groups. In Subsection 8.1, we state the main result of this section and explain how it can be reduced to finding a suitable 1-form that satisfies a certain boundedness condition; this boundedness condition can be thought of as a discretised version of being a primitive of a bounded 2-form. In Subsection 8.2, we will provide a linear representation of the filiform Lie group in all dimensions and construct an exact invariant 2-form from it. In Subsection 8.3, we will show how to modify this 2-form to obtain a suitable exact bounded 2-form. Finally, in Subsection 8.4, we will show that this bounded 2-form is the differential of a 1-form that satisfies the boundedness condition from Subsection 8.1 and deduce the desired lower bounds on the Dehn function of , .
Before going into the proof of Theorem 8.1, we summarise our approach for obtaining the desired lower bound on +1, +1 . It suffices to find a family of null-homotopic words +1, of length ≃ and area ≃ . A natural candidate for +1, is the word Ω +1 ( ) ∶= Ω +1 ( , … , ) defined via the embedding of +1 in the first factor of +1, +1 . The reason for this is that its image with respect to the projection +1, +1 → is a product of the words − 1 Ω ( ) −1 1 and Ω ( ), which both have area in . One way of showing that these two words have the asserted with the commutator bracket [ , ] ∶= − on matrices. Thus, we can obtain a linear representation of by computing the image exp( ). We will now make this explicit. For this we introduce the notation and observe that for = ( 1 , … , ) and ∈ we obtain for a suitable ∈ −1 . Moreover, it is easy to derive by induction that Proof. The first identity is an immediate consequence of the following identities To derive the identities for 2 ⩽ ⩽ denote by ∈ −1 the th unit vector and observe that We deduce that in local coordinates we have , * ,0 = ∑ − has left -invariant differential In fact, is an explicit realisation in the coordinates of the 2-form 1 ∧ from the proof of Proposition 7.11. The reason we consider it is that it defines a p-central extension of . However, as we have seen we face the problem that this form does not survive in 2 ( , , ) for < . Thus, we cannot use it directly to obtain a lower bound on , ( ) by defining a p-central extension and, as we have shown, there is actually not even a ( − 1)-central extension of , for odd and < . To overcome this problem and confirm our intuition that , ( ) ≽ −1 , we will now pursue the approach sketched in Subsection 2.2 of constructing a suitable perturbation 0 of with bounded differential, which has integral ≃ on certain ( + 1)-fold iterated commutators in ; they arise as images of null-homotopic words in +1, +1 with respect to the canonical projection. In view of Subsection 8.1 this will allow us to deduce the desired lower bounds on the Dehn function.
Remark 8.5. One checks by induction on that the exponent sum of 1 in any prefix word of Ω ( ) ±1 lies in the interval [−( − 1) , 0], for all ∈ ℕ.
We will show that the integral of the form along the loop defined by Ω ( ) in is . This is one way to prove that the null-homotopic words Ω ( ) are area maximising in . It makes a good candidate for showing that Ω +1 ( ) = [ 1 , Ω ( )] also has area ≃ in . However, and thus the integral of along Ω +1 ( ) vanishes (this is a direct consequence of (8.5) and the left -invariance of ). This means that the form would not allow us to obtain the desired lower bounds on the Dehn function.
We will show that we can avoid this problem by replacing by a continuous perturbation 0 with the property that the differential 0 exists for 1 ≠ 0 and coincides with for 1 > 0 and with − for 1 < 0. Moreover, to simplify our calculations, we will consider the null-homotopic word +1, = Its projection to consists of two disjoint loops + and − , and a line connecting their basepoints. By Remark 8.5, the exponent sum of 1 in any prefix word of + (respectively, − ) is positive (respectively, negative). In particular, the image of + is contained in the set, where 0 = , while the image of − is contained in the set, where 0 = − . Since is the 2-form defining the central extension +1 → , one can deduce from this that ∫ +1, 0 ≃ . Below we provide the details of this argument and calculate the precise value of ∫ +1, 0 . We start by defining 0 : A direct calculation shows To evaluate the integral of 0 along +1, we need to evaluate it along each part of the loop. For this we will use the following result: (1) ∫ 1 0 = 0 and ′ 1 = 1 + for ′ ∈ with ′ = 1 ( ); (2) ∫ 2 0 = sgn( ) −1 ( −1)! and ′ 1 = 1 for ′ ∈ with ′ = 2 ( ).
Proof. Assertion (1) follows from Lemma 8.3, the vanishing of 0, on 1 , and For Assertion (2) we first observe that We deduce thaṫ2 and that the 1 -coordinate is constant along 2 ( ). Thus, where in (1) we use the binomial formula 0 = (1 + (−1)) −1 = ∑ −1 =0 (−1) . This completes the proof. □ For a word ( 1 , 2 ), we introduce the notation 1 ( ) for its 1 -exponent sum. Lemma 8.6 shows that (1) if a word ( 1 , 2 ) represents the element in for ∈ then 1 coincides with 1 ( ); (2) we can compute ∫ +1, 0 by reading +1, from left to right and adding a contribution for every power of 2 that we encounter. The contribution of such an 2 -power will depend solely on the 1 -exponent sum of its prefix word and the numerical value of the exponent of this 2power. In particular, this essentially reduces the computation of ∫  In particular, it is immediate from the fact that the exponent signs of the ± 2 are alternating in Ω ( ) that the same holds for Ω +1 ( ) and it follows readily that (1) holds for Ω +1 ( ).
Since the area of loops of length ( ) ⩽ Proposition 9.2 (Cornulier). Let and ′ be homogeneous metric spaces. If there exists a ( )-SBE → ′ , then for any nonprincipal ultrafilter and sequence of scaling factors ( ) the metric spaces Cone ( , ) and Cone ( ′ , ) are bi-Lipschitz homeomorphic.
In particular, if a homogeneous space is ( )-SBE to a self-similar homogeneous space ′ , then the latter is the asymptotic cone of up to bi-Lipschitz homeomorphism. Not all simply connected nilpotent Lie groups admit left-invariant self-similar proper geodesic metrics, only the Carnot gradable ones do.  [22]. Cornulier's input in [11] is in the improvement of in terms of finer invariants of the structure of . We will give low-dimensional examples in Table 4. [9,30,31]). Let and ′ be two simply connected nilpotent Lie groups. The following are equivalent. Proof of Corollary 9.5. Assuming (i), we deduce (ii) from Theorem 1.2. (ii) implies (iii) by Theorem 9.3. Finally, (iii) implies (i) by Proposition 9.2. □ Remark 9.6. Corollary 9.5 holds for locally compact groups with polynomial growth , where the construction of ( ) requires additional steps. In particular, one first has to pass to a nilshadow of the Lie shadow of , see Breuillard [4].

Corollary 9.5 (Pansu and Cornulier
Corollary 9.5 leaves the problem of evaluating the range of such that a given pair of groups with identical asymptotic cones can be ( )-equivalent. The question was raised by Cornulier [11,Question 6.20]. For the pair ( × −2 , , −1 ), our Theorem C states that one must have ⩾ 1∕ (2 ), which for the first case of interest = 4 implies ⩾ 1∕8. These are the first examples for which a positive lower bound is known. We will prove Theorem C at the end of this section.

Large-scale fillings and SBE
Our main tool for proving Theorem C is the following technical lemma. is a filling pair for .
Before starting the proof we fix some conventions and notations. We will fix Cayley graphs of and ′ , and a loop in will be a loop in the Cayley graph of (not necessarily based at 1). When we speak of maps to (respectively, ′ ) we will from now on mean maps to their respective Cayley graphs.
A combinatorial disk Δ ∶= ( , ) filling a loop is defined by the following data: a CW-complex structure on the closed 2-dimensional unit ball with 2-cells Δ 1 , … , Δ and injective attaching maps in all dimensions, and a continuous map ∶ (1) → from the 1-skeleton of to the Cayley graph of , such that parametrises | Δ and maps vertices to vertices. We will denote ∶= | Δ the boundary loops of the 2-cells and say that Δ is a filling of by loops 1 , … , .
Retaining the above notation, one can check that admits ( , ) as a filling pair if and only if there is a constant 0 > 0 such that every loop of length ⩽ based at the identity in admits a filling by a combinatorial disk such that ≲ , ( (1) ) is contained in a ball of diameter ≲ around the origin and parametrises a loop of length ⩽ 0 . This is straightforward and wellknown for Dehn functions and generalises readily to filling pairs.
We deduce that the Dehn function of has to satisfy ≲ (1+ )( −1)+ (1+ ) + (1+ ) 2 + ( −1) . This yields the inequality A straightforward calculation shows that for = 1 2 this inequality is not satisfied. Since both of the terms on the right are increasing functions in ∈ [0, 1) the inequality cannot be satisfied for any ∈ [0, 1 2 ], yielding the desired lower bound. □

OVERVIEW IN LOW DIMENSIONS
In this section, we provide a complete overview of the real nilpotent Lie algebras of dimension less or equal to 6 together with the best estimates that we can find on their Dehn functions. By the Dehn function (respectively, the centralised Dehn function) of a Lie algebra , denoted ( ) respectively, cent ( ), we mean the Dehn function (respectively, the centralised Dehn function) of its associated simply connected nilpotent Lie group (that is, Lie( ) = ). A complete classification of real nilpotent Lie algebras of dimension ⩽ 6 was given by de Graaf [15]. We will use his notation † L , , where is the dimension and is an integer. Note that in dimension ⩽ 5 all Dehn functions were computed by Pittet [34]. We still list them for the sake of completeness. We list the nilpotent Lie algebras together with their structure, their homogeneous dimension and the best-known estimates on their Dehn functions in Tables 1-4. Table 1 contains all nilpotent Lie algebras of dimension at most 5 and Tables 2-4 those of dimension 6 ordered by their nilpotency classes and homogeneous dimension hdim( ) ∶= ∑ ⩾1 dim ∕ +1 . The latter is a quasi-isometry invariant, as it coincides with the exponent of growth of the corresponding group [25, Theorem II.1].
We will now give some explanations regarding the contents of our tables. In dimension 6, we do not list decomposable Lie algebras (that is, Lie algebras that split as a direct product of lower dimensional ones) except if their class of Lie algebras with the same Carnot graded algebra consists of more than one element; this is to keep our tables as compact as possible. More generally, we group Lie algebras by their associated Carnot graded algebras, starting with the unique one that is Carnot. The nonzero brackets defining the structure of the respective Lie algebras are provided in an abbreviated form: for instance, the notation 12 = 34 = 5 means that [ 1 , 2 ] = [ 3 , 4 ] = 5 and defines the 5-dimensional Heisenberg algebra.
In most cases, our estimates on ( ) are derived as follows.
(1) The upper bound is given by the universal upper bound of +1 on the Dehn function of a nilpotent group of nilpotency class [21]. (2) The lower bound is given by the centralised Dehn function cent ( ). It is obtained by providing a suitable central extension of maximal distortion.
For (2) we provide a maximally distorted central extension in abbreviated form in the table. Let us illustrate this via the example of L 5,5 . In this case we claim that a central extension of maximal † Note that de Graaf's precise notation is , rather than L , . F I G U R E 5 Nilpotent Lie algebras of dimension ⩽ 5 and how they are related. The notation ⟶ means that is a -central extension of by (see Tables 1-4 for the explicit extensions) and gr ⤏ means that = gr( ). distortion is given by = 14 = 35. This is short-hand for the extension → e → , where satisfies = [ ( 1 ), ( 4 )] = [ ( 3 ), ( 5 )] for any section ∶ → e. Verifying the existence of this extension is easy via the well-known identification of central extensions with second cohomology classes given by Proposition 7.2. Indeed, in the case of L 5,5 the extension = 14 = 35 corresponds to the 2-form ∶= 1 ∧ 4 + 3 ∧ 5 , where 1 , … , 5 is a dual basis of the basis 1 , … , 5 . We readily deduce from the structure of L 5,5 that 1 = 2 = 5 = 0, 3 = − 1 ∧ 2 and 4 = − 1 ∧ 3 − 2 ∧ 5 . Thus, we obtain that = 0 and that defines a non-trivial cohomology class. For the cases where there are either better estimates on the Dehn function than one can obtain from the above method or where estimates are well-known we provide a reference to the literature or previous sections. Finally, note that the Dehn functions of the decomposable algebras that we did not list can easily be deduced from Lemma 3.9 and the Dehn functions of their factors.
Remark 10.1. We indicate all relations via central extensions between nilpotent Lie algebras of dimension ⩽ 5 in Figure 5; if is 5-dimensional we also provide at least one 6-dimensional central extension.
Note that there are a total of 5 cases for which we were not able to determine the precise Dehn functions via any method. In particular, in these cases the bounds from (1) and (2) do not match. We summarise the state of the art for these cases. Proof. For the existence of a -central extensions we refer to the concrete -central extensions indicated in the tables with the arguments being the same as the ones given above. The proof of the non-existence of a ( + 1)-central extension is by performing computations similar to the ones in Subsection 7.4. Note that for the Carnot case the computations are more elegant than for the non-Carnot case, since the differential preserves the grading. The only non-Carnot Lie algebra in  is L 6,14 ; the corresponding computation is more cumbersome but no harder.
Rather than giving details for all cases, we will restrict to the concrete example of the Carnot Lie algebra L 6,20 and leave the remainder of the computations as an exercise to the reader. By definition L 6,20 is 3-step nilpotent.
To show that there is no 4-central extension it suffices to prove that 2 ( , ) 4 = 0. Recall that L 6,20 is defined by the generating set { 1 , … , 6 } and the following nonzero relations We denote its dual basis { 1 , … , 6 } and, as before, we use the notation , = ∧ , and so on.

QUESTIONS AND SPECULATIONS
We start with a question whose answer would complete the computation of the Dehn functions of all simply connected nilpotent Lie groups of dimension less or equal 6. With the exception of L 6,14 all groups corresponding to the Lie algebras in  are possible candidates for a positive answer to the following question. Question 11.2. Does there exist a Carnot gradable simply connected nilpotent Lie group such that its Dehn function and its centralised Dehn function both grow like , but with different exponents ?
More generally, we might expect a general picture for Dehn functions of central products. Let and be nilpotent Lie algebras of step , respectively, , with ⩾ ⩾ 2, and 1-dimensional centres and ′ . Let ∶ → ′ be an isomorphism between their centres and let × be their central product. We denote by , and ∶= × the associated simply connected Lie groups. We explain the intuition behind this conjecture. First we observe that the fact that the centres in consideration are 1-dimensional implies that there is still a cocycle , defining the -central extension → ∕ . As for our examples , , this cocycle represents the 'trivial' central extension × → × . It is thus -central and, in particular, it will only be -central if = . Moreover, there is no -central extension for ⩾ + 1 (see Lemmas 7.9 and 7.10). Hence, we can at best hope for a lower bound of on the Dehn function of by using central extensions. On the other hand, we can in general not even hope for this, as for > the form does not provide such an extension and our examples show that no other -central extension might exist. However, it seems reasonable to believe that perturbation arguments similar to the ones developed in Section 8 can be used to show that the Dehn function of is ≽ . This explains our guess for the lower bound.
For the upper bound the key intuition is that it should still be possible to commute central words ( ) in the generators of with arbitrary words ( ) at a lower cost than +1 by using what we will now call the 'central word trick': one replaces ( ) by a suitable word ′ ( ′ ) in the generators ′ of at cost ≺ +1 and then exploits that [ , ′ ] = 1 to commute it with . For the overall approach one should mimic the boot-strapping trick of using an inductive argument on the nilpotency class that we applied in Section 6 (also see its sketch in the second half of Subsection 2.2).
The basic idea would be to first reduce the word ( ) to a word ( ), where the letters live in a subgroup < of nilpotency class strictly lower than (in our case, = , while = −1 ). Such a will presumably have length 2 . We then assume that the conjecture holds by induction for − 1 and apply it in the central product × to commute ( ) with other words in at cost ≺ ⋅ = +1 . As we saw in Section 6 this simple trick, used in the right way, is the fundamental reason why our argument works.
Once we inductively reduced to a 2-step nilpotent central product, we can invoke Olshanskii and Sapir's result that the Dehn function of such a group is bounded above by 2 log( ) [28]. This would allow us to conclude. We remark that while they do not say this explicitly, the reason why Olshanskii and Sapir's argument for 2-step nilpotent groups works ultimately also boils down to the central word trick (and we are convinced that the authors were aware of this). However, as we have seen in Section 6 it is far from obvious how to make such an argument work in higher step. There are various reasons for this, for instance, to mention just one of them, making it work requires the reduction step that turns words of length in into words of length 2 in a suitable alphabet at sufficiently low cost, a step that was not needed for 2-step nilpotent groups.
The fact that already for the specific class of groups , −1 , whose structure is as simple as one may hope for, the argument turns out to be highly technical, suggests that actually proving Conjecture 11.3 in general will at the very least require the development of a refined version of our methods and potentially even a completely different approach.
Finally, it is worth noting that it would even be interesting to prove Conjecture 11.3 for other specific classes of examples. Indeed, well-chosen classes of examples might well produce new groups that satisfy all the main conclusions of our results in the introduction. A first such class to consider would be the general class of groups , for which so far we were only able to determine the precise Dehn function for ∈ {2, − 1, } (see also Remark 8.13).
Considering specific classes of examples seems particularly tempting, because, with some real speculation involved, a well-chosen class of examples could potentially produce nilpotent groups with Dehn functions strictly between and +1 for all integers ⩾ 3, generalising Wenger's examples [40], or, on an even more speculative note, even nilpotent groups whose Dehn functions do not have integer exponents.

A C K N O W L E D G E M E N T S
We are grateful to Yves Cornulier and Christophe Pittet for helpful comments on a previous version of this paper. We also thank Francesca Tripaldi for helpful discussions. We thank the anonymous referee whose comments led to significant improvements in the exposition of our work and results. Claudio Llosa Isenrich was supported by a public grant as part of the FMJH, by the Max Planck Institute for Mathematics and by the Lise Meitner fellowship M2811-N of the Austrian Science Fund (FWF). Gabriel Pallier was supported by the European Research Council (ERC Starting Grant 713998 GeoMeG 'Geometry of Metric Groups').

J O U R N A L I N F O R M AT I O N
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