Hyperbolicity in non-metric cubical small-cancellation

Given a non-positively curved cube complex $X$, we prove that the quotient of $\pi_1X$ defined by a cubical presentation $\langle X\mid Y_1,\dots, Y_s\rangle$ satisfying sufficient non-metric cubical small-cancellation conditions is hyperbolic provided that $\pi_1X$ is hyperbolic. This generalises the fact that finitely presented classical $C(7)$ small-cancellation groups are hyperbolic.


Introduction
A cubical presentation is a higher dimensional generalisation of a classical group presentation in terms of generators and relators.A non-positively curved cube complex X plays the role of the "generators", and the "relators" are local isometries of non-positively curved cube complexes Y i → X.The associated group is the quotient of π 1 X by the normal closure ⟨⟨ {π 1 Y i } ⟩⟩ π 1 X of π 1 Y i .As in the classical setting, this group is the fundamental group of X with the Y i 's coned off.Likewise, cubical small-cancellation theory, introduced in [Wis21], is a generalisation of classical small-cancellation theory (see e.g.[LS77]).In both the classical and cubical cases, the small-cancellation conditions are expressed in terms of pieces.A piece in a classical presentation is a word that appears in two different places among the relators.The non-metric small-cancellation condition C(p) where p > 1 asserts that no relator is a concatenation of fewer than p pieces.The metric smallcancellation condition C ′ (α) α ∈ (0, 1) asserts that |P | < α|R| whenever P is a piece in a relator R. Note that C ′ ( 1 p ) ⇒ C(p + 1).Pieces in cubical presentation are defined similarly, and the same implication holds in the cubical case.
Cubical small-cancellation has proven to be a fruitful tool in the study of groups acting on CAT(0) cube complexes.It was used by Wise as a step in his proof of the Malnormal Special Quotient Theorem [Wis21], and as such, played a crucial role in the proofs of the Virtual Haken and Virtual Fibering conjectures [Ago13].Cubical presentations and cubical small-cancellation theory were also studied and utilised in [Are23a, Are23b, AH22, FW16, HW22, Jan20, JW22, PW18].While classical small cancellation groups have virtual cohomological dimension ≤ 2 [Lyn66], there exist cubical small cancellation groups with arbitrarily large virtual cohomological dimension, which is moreover controlled by cd π 1 X and cd π 1 Y i [Are].
To illustrate the difference between metric and non-metric conditions, consider the following presentation: ⟨a, b | a n w⟩.When w is a long messy word (read: small cancellation) starting and ending in b, then the C(p) condition holds for all n.However a n−1 is a piece!So C ′ (α) fails for sufficiently large n.Similar examples can be produced in the cubical setting.For instance, let X = S ∨ A where S is a cubulated surface and A is a circle.Let w be a small-cancellation path in X whose initial and terminal edges lie in S. Let w be the lift of w to X, and let W be the combinatorial convex hull of w.Let a n be a length n arc that immerses onto A. Let Y be the quotient of a n ∪ W , identifying the endpoints of a n and w.Then ⟨X | Y ⟩ is a cubical presentation satisfying C(p), but not C ′ ( 1 p ), when n ≫ 0. The pumping lemma shows that for any X with π 1 X non-elementary hyperbolic, there are presentations X * that are non-metric small cancellation, but not metric small cancellation.
It is a fundamental result of classical small-cancellation theory that a group admitting a finite presentation satisfying the classical C ′ ( 1 6 ) or C(7) condition is hyperbolic.In analogy with the metric classical small-cancellation case, a cubical C ′ ( 1 14 ) presentation ⟨X | Y 1 , . . ., Y s ⟩ yields a hyperbolic group if π 1 X is hyperbolic and the Y i are compact [Wis21,Thm 4.7].However, the proof of that result does not extend to the non-metric case.The goal of this paper is to prove the following statement which recovers and generalises the result from the C ′ ( 1 14 ) setting.
1.1.Proof strategy.The main ingredients of the proof are the notion of the piece metric (Definition 4.1) and Papasoglu's thin bigon criterion for hyperbolicity (Proposition 3.1).
The most immediate way of proving hyperbolicity for finitely presented C ′ ( 1 6 ) groups is to show that a linear isoperimetric inequality holds for their Cayley complexes.This follows from the fact that C ′ ( 1 6 ) presentations satisfy Dehn's algorithm by Greendlinger's Lemma (see for instance [LS77, V.4.5]).In the C(7) setting, one is no longer guaranteed to have a Dehn presentation (consider the ⟨a, b|a n w⟩ examples described above).Instead, the usual way of proving hyperbolicity in this generality relies on the combinatorial Gauss-Bonnet Theorem.Another way is to realise that reduced disc diagrams satisfy C ′ ( 1 6 ) if we regard all pieces as having length 1.In fact, this viewpoint leads to the piece-metric.
To illustrate the basic idea behind our strategy, we sketch a proof of hyperbolicity in the C(7) case using the piece-metric and the thin bigon criterion.The definition of the classical C(n) condition and of all the diagrammatic notions introduced for cubical presentations in Section 2.2 can be particularised to this setting, and a version of Greendlinger's Lemma also holds (see for instance [LS77, V.4.5]).
Illustrative theorem.Let X be a 2-complex satisfying the C(7) small-cancellation condition.Then X is hyperbolic with the piece metric.
This implies hyperbolicity of finitely presented C(7) groups, since in that case the piece-metric and the usual combinatorial metric on the Cayley graph are quasiisometric (see Proposition 4.3).
Proof.We check that all bigons in X (0) are ϵ-thin in the piece metric for some ϵ > 0, and apply Proposition 3.1 to conclude that X is hyperbolic.
Let γ 1 , γ 2 be piece-geodesics forming a bigon in X, and let D → X be a reduced disc diagram with ∂D = γ 1 γ2 .We claim that D is a (possibly degenerate) ladder, and hence that the bigon γ 1 , γ 2 is 1-thin, since by definition any two cells in a ladder intersect along at most 1 piece.
Indeed, by Greendlinger's Lemma, D is either a ladder, or contains at least 3 shells and/or spurs.First note that D cannot have spurs, as these can be removed to obtain paths γ ′ 1 , γ ′ 2 with the same endpoints as γ 1 , γ 2 , and which are shorter in the piece metric, thus contradicting that γ 1 , γ 2 are piece-geodesics.
If D has at least 3 shells, then at least 1 shell S must have its outerpath Q contained in either γ 1 or γ 2 .Since both cases are analogous, assume Q ⊂ γ 1 , and let R be the innerpath of S. Since S is a shell and X satisfies C(7), then R is the concatenation of at most 3 pieces, so |R| p < |Q| p , and the path γ ′ 1 obtained from γ 1 by traversing R instead of Q is the concatenation of less pieces than γ 1 , contradicting that γ 1 is a piece-geodesic.
Thus, D is a ladder, and the proof is complete.□ 1.2.Structure of the paper.The paper is organised as follows.In Section 2, we give background on cube complexes, cubical group presentations, and cubical small-cancellation.In Section 3, we recall a criterion for hyperbolicity for groups acting on graphs.In Section 4, we define and analyse the piece metric.In Section 5, we prove Theorem 5.1.Lea13,Wis21].A non-positively curved cube complex is a cell-complex X whose universal cover X is a CAT(0) cube complex.A hyperplane H in X is a subspace whose intersection with each n-cube [0, 1] n is either empty or consists of the subspace where exactly one coordinate is restricted to 1 2 .For a hyperplane H of X, we let N ( H) denote its carrier, which is the union of all closed cubes intersecting H.The combinatorial metric d on the 0-skeleton of a non-positively curved cube complex X is a length metric where the distance between two points is the length of the shortest combinatorial path connecting them.A map ϕ : Y → X between non-positively curved cube complexes is a local isometry if ϕ is locally injective, ϕ maps open cubes homeomorphically to open cubes, and whenever a, b are concatenable edges of Y , if ϕ(a)ϕ(b) is a subpath of the attaching map of a 2-cube of X, then ab is a subpath of a 2-cube in Y .
2.2.Cubical presentations.We recall the notion of a cubical presentation, and the cubical small-cancellation conditions from [Wis21].
A cubical presentation ⟨X | Y 1 , . . ., Y m ⟩ consists of a non-positively curved cube complex X, and a set of local isometries Y i ↬ X of non-positively curved cube complexes.We use the notation X * for the cubical presentation above.As a topological space, X * consists of X with a cone on Y i attached to X for each i.The vertices of the cones on Y i 's will be referred to as cone-vertices of X * .The cellular structure of X * consists of all the original cubes of X, and the "pyramids" over cubes in Y i with a cone-vertex for the apex.
As mentioned in the introduction, cubical presentations generalise classical group presentations.Indeed, a classical presentation complex associated with a group presentation G = ⟨S | R⟩ can be viewed as a cubical presentation where the nonpositively curved cube complex X is just a wedge of circles, one corresponding to each generator in S. The complexes Y i correspond to relators r i in R. Each cycle Y i has length |r i |, and the local isometry Y i ↬ X is defined by labelling the edges of Y i with the letters of r i .
The universal cover X * consists of a cube complex X with cones over copies of Y i 's.The complex X is a covering space of X.A combinatorial geodesic in X * is a combinatorial geodesic in X, viewed as a path in X * .2.3.Disc diagrams in X * .Throughout this paper, we will be analysing properties of disc diagrams, which we introduce below together with some associated terminology: A map f : X −→ Y between 2-complexes is combinatorial if it maps cells to cells of the same dimension.A complex is combinatorial if all attaching maps are combinatorial, possibly after subdividing the cells.
A disc diagram is a compact, contractible 2-complex D with a fixed planar embedding D ⊆ S 2 .The embedding D → S 2 induces a cell structure on S 2 , consisting of the 2-cells of D together with an additional 2-cell, which is the 2-cell We emphasize that this definition differs slightly from the definition of a conecell in the literature, where A is simply a circle; allowing A to be an arbitrary annular diagram, D implicitly comes equipped with a choice of cone-cells.
The square part D □ of D is a subdiagram which is the union of all the squares that are not contained in cone-cells.
A square disc diagram is a disc diagram whose square part is the whole diagram, i.e. it contains no cone-cells.A mid-interval in a square, viewed as A dual curve in a square disc diagram D is a curve which intersect each closed square either trivially, or along a mid-interval, i.e., a dual curve is a restriction of a hyperplane in X to D. We note that for each 1-cube of D, there exists a unique dual curve crossing it [Wis21,2e].
The complexity of a disc diagram D in X * is defined as We say that D has minimal complexity if Comp(D) is minimal in the lexicographical order among disc diagrams with the same boundary path as • weakly reduced if no moves (1) -(5) from Definition 2.1 can be performed in D.
Note that if D has minimal complexity then D is reduced, and that, in particular, each reduction move outputs a diagram D ′ with Area(D ′ ) < Area(D) and ∂D ′ = ∂D.Consequently: (1) ∂D = ∂D ′ = ∂D ′′ , (2) D ′ is weakly reduced and D ′′ is reduced, (3) D ′ is obtained from D after a a finite number of moves of types (1)-( 5), and D ′′ is obtained from D after a finite number of moves of type (0)-(5).
Remark 2.4.Many theorems about disc diagrams in the literature assume that the disc diagram is reduced or minimal complexity, but it is in fact sufficient to consider weakly reduced diagrams.For example, this is the case with Lemma 2.7 (the Cubical Greendlinger's Lemma).
2.4.Cubical small-cancellation.We use the convention where ρ denotes the path ρ with the opposite orientation.A grid is a square disc diagram isometric to the product of two intervals.Let ρ and η be two combinatorial paths in X * .We say ρ and σ are parallel if there exists a grid E → X * with ∂E = µρνη, where the dual curves dual to edges of ρ, ordered with respect to its orientation, are also dual to edges of η, ordered with respect to its orientation.
and h(e i ) and h(f i ) are the curves dual to e i and f i respectively, then ρ is a piece if h(e i ) = h(f i ) for each i ∈ {1, . . ., k}.
, where Y i is a fixed elevation of Y i to the universal cover X, and either i ̸ = j or where , where H is a hyperplane that is disjoint from Y i .Each abstract contiguous wall-piece P induces a map P → Y i , and a contiguous wall-piece of Y i is a combinatorial path ρ → P in an abstract contiguous wall-piece of Y i .A piece is a path parallel to a contiguous cone-piece or wall-piece.The difference between contiguous pieces and pieces is illustrated in Figure 3.
For an integer p > 0, we say X * satisfies the C(p) small-cancellation condition if no essential combinatorial closed path in Y i can be expressed as a concatenation of less than p pieces.For a constant α > 0, we say X * satisfies the C ′ (α) smallcancellation condition if diam(P ) < α∥Y i ∥ for every piece P involving Y i .A ladder is a disc diagram (D, ∂D) → (X * , X (0) ) which is an alternating union of cone-cells and/or vertices C 0 , C 2 . . ., C 2n and (possibly degenerate) pseudo-grids E 1 , E 3 . . ., E 2n−1 , with n ≥ 0, in the following sense: (1) the boundary path ∂D is a concatenation λ 1 λ 2 where the initial points of λ 1 , λ 2 lie in C 0 , and the terminal points of λ 1 , λ 2 lie in C 2n , (2) (3) the boundary path ∂C i = ν i−1 α i µ i+1 β i for some ν i−1 and µ i+1 (where ν −1 and µ 2n+1 are trivial), and (4) the boundary path Lemma 2.7 (Cubical Greendlinger's Lemma [Wis21,Jan20]).Let X * = ⟨X | Y 1 , . . ., Y s ⟩ be a cubical presentation satisfying the C(9) condition, and let D → X * be a weakly reduced disc diagram.Then one of the following holds: • D is a ladder, or • D has at least three shells of degree ≤ 4 and/or corners and/or spurs.
We note that our definition of ladder differs slightly from the definitions in [Wis21, Jan20], so that a single cone-cell and a single vertex count as ladders here.Also, the statements in [Wis21,Jan20] assume that the disc diagrams are reduced/minimal complexity, but the proofs work for weakly reduced disc diagrams.

Hyperbolic background
We explain the convention we will follow.A pair (Y, d) is a metric graph, if there exists a graph Γ such that Y is the vertex set of Γ, and d is defined as follows.For each edge of Γ, we assign a positive number which is the length of that edge.The length of a simple path in Γ is the sum of the lengths of the edges in the path.A metric d on a set Y is a graph metric, if (Y, d) is a metric graph.
Of course, the converse also holds.

The piece metric
Let X * = ⟨X | Y 1 , . . ., Y s ⟩ be a cubical presentation.As explained in Section 2.2, we write X * to denote the complex X with cones over Y i 's attached.In particular, X can be viewed as a subspace of X * .The preimage of X in the universal cover X * of X * is denoted by X.Note that X is a covering space of X.The preimage of the 0-skeleton of X in X * is also the 0-skeleton of X, so it is denoted by X (0) .Definition 4.1.The piece length of a combinatorial path γ in X (0) is the smallest n such that γ = ν 1 • • • ν n where each ν k is a 1-cube or a piece.The piece metric d p on X (0) is defined as d p (a, b) = n where n is the smallest piece length of a path from a to b.
We note that d p is a graph metric when X (0) is viewed as the graph with all edges of length 1 obtained from the 1-skeleton X (1) of X by adding extra edges between vertices contained in a single piece.We will denote this graph by ( X (0) , d p ).
A piece decomposition of a path γ is an expression γ = ν 1 • • • ν k , where each ν i is a piece or 1-cube.We make the following easy observation: Lemma 4.2.Let γ, γ 1 , γ 2 be piece-metric geodesics in X (0) where γ = γ 1 γ 2 .Then Proof.Any piece decomposition γ = ν 1 • • • ν k yields piece decompositions of both γ 1 and γ 2 , where at most one piece ν i for i ∈ {1, . . ., k} further decomposes into the concatenation of 2 pieces ν ′ i , ν ′′ i , so Similarly, any two piece decompositions of γ 1 and γ 2 can be concatenated to obtain a piece decomposition of γ. □ We now prove a few basic facts about the piece metric.First, it is quasi-isometric to the combinatorial metric under fairly weak hypotheses.Since there are only finitely many Y i 's and each Y i is compact, there must be an upper bound on the diameter of a simple essential curve in Y i with respect to d and thus with respect to d p , which implies the second statement.□ Corollary 4.4.Suppose that π 1 X is hyperbolic.Let D → X * be a square diagram with boundary ∂D = γλ where γ is a d p -geodesic, and λ is a d-geodesic.Then the bigon D is M -thin for a uniform constant M .
Proof.First note that D → X * is a square diagram in X, but it also lifts to X.The metric d p also lifts to X (0) , and by Proposition 4.3 d, d p are quasi-isometric on X (0) , and therefore on X (0) .The statement then follows from the uniform bound on the Hausdorff distance between geodesics and quasi-geodesics in hyperbolic spaces.□ We note that ladders are thin with respect to the piece metric.
Proposition 4.5.Suppose that π 1 X is hyperbolic.Let D → X * be a ladder with boundary ∂D = λ 1 λ 2 as in Definition 2.6 where each subpath of λ i contained in a single pseudo-grid is a geodesic.Then the bigon λ 1 , λ 2 is ϵ-thin with respect to d p for a uniform constant ϵ > 0 dependent only on X * .
Proof.We only show that λ 1 ⊆ N ϵ (λ 2 ), since the argument for λ 2 ⊆ N ϵ (λ 1 ) is analogous.Let x ∈ λ 1 .We want to show that d p (x, λ 2 ) ≤ ϵ.If x belongs to a cone-cell C, then by the definition of the ladder, λ 2 also intersects C, so d p (x, λ 2 ) is bounded by the piece-metric diameter of C, which is uniformly bounded by some constant ϵ 1 by Proposition 4.3.Otherwise x lies in a pseudo-grid.Let ρ, η be subpaths of λ 1 , λ 2 respectively, contained in the pseudo-grid which contains x.The paths ρ, η are both combinatorial geodesics by the assumption.By Proposition 4.3 ρ, η start and end at a uniform distance, since they lie in the same cone-cell.By hyperbolicity of X, there exists ϵ 2 > 0 such that ρ, η ϵ 2 -fellow travel.The conclusion follows with ϵ = max{ϵ 1 , ϵ 2 }. □ In the proof of Theorem 5.1 we will use the following technical lemma.ℓQr is naturally paired with an edge of γ.For every piece ν in γ, we consider all the dual curves h 1 , . . ., h n starting at ν that exit E in ℓQr.
These define a collection of edges in ℓQr, and every subcollection of such consecutive edges forms a path that is a piece, as it is parallel to some path contained in one of Y i .By grouping consecutive edges into maximal subpaths contained in one of ℓ, Q, or r, we get pieces ν 1 , . . ., ν k whose interiors are pairwise disjoint (ordered consistently with the orientation of ℓQr), and say that ν projects to ν 1 , . . ., ν k .
First we claim that each of ℓ, Q, r contains at most one piece ν i .Suppose to the contrary that ν i , ν i+1 are both contained in ℓ (and the same argument applies to Q, r).Then each dual curve starting at an edge of ℓ lying between ν i and ν i+1 must intersect at least one dual curve starting at edges of ν i , ν i+1 , as otherwise it would also lie in a projection of ν, yielding a cornsquare in ℓ.
Thus we can denote the projection of ν by ν ℓ , ν Q , ν r where each piece is a possibly empty projection onto ℓ, Q, r respectively.See left diagram in Figure 5.We will assume that they are oriented consistently with ℓ, Q, r respectively, not necessarily consistently with ν.
be the induced piece-decomposition where we only write nontrivial pieces.In particular, i ℓ : {1, . . ., n ℓ } → {1, . . ., n} is an injective function.We now claim that i ℓ is monotone.Suppose to the contrary, that 1 ≤ j < k ≤ n ℓ but i ℓ (j) > i ℓ (k).Then there must exists a cornsquare in the connected subpath of ℓ containing ν ℓ i ℓ (k) and ν ℓ i ℓ (j) , which is a contradiction.Analogously, we get , and the functions i Q , i r are monotone.These are not necessarily the minimal piece decompositions, but certainly we have |ℓ| p ≤ n ℓ , |Q| p ≤ n Q , and |r| p ≤ n r .To prove the lemma we will show that |Q| p + n ℓ + n r ≤ n + 3.
Note that i ℓ (n ℓ ) is the largest index in {1, . . ., n} such that ν i ℓ (n ℓ ) has non-trivial projection onto ℓ, and similarly i r (1) is the lowest index in {1, . . ., n} such that ν ir(1) has non-trivial projection onto r.See middle diagram in Figure 5. Since i ℓ , i r are monotone, n ℓ ≤ i ℓ (n ℓ ) and n r ≤ n − i r (1).Thus, it remains to prove that is a single piece in Q.Indeed, the dual curves starting in must all intersect a dual curve starting in ν i ℓ (n ℓ ) and exiting the diagram in ℓ.See right diagram in Figure 5. Similarly, let k r be the smallest number such that i Q (k r ) > i r (1) and note that ν This proves that |Q| p ≤ i r (1) − i ℓ (n ℓ ) + 3 and completes the proof.□

Proof of hyperbolicity
In the proof of the next theorem, we show that, under suitable assumptions, ( X (0) , d p ) is a δ-hyperbolic graph to deduce that π 1 X * is hyperbolic.The basic strategy is similar to [Wis21, Thm 4.7], but the details in this case are significantly more involved.
Before we proceed with the proof of the above theorem we introduce a construction that is used in the proof.Let Y ⊂ X.The cubical convex hull of Y in X is the smallest cubically convex subcomplex of X contained in Y .That is, it is the smallest subcomplex Hull(Y ) satisfying that whenever a corner of an n-cube c with n ≥ 2 lies in Hull(Y ), then c ⊂ Hull(Y ).
where for each i = 0, . . ., n − 1 the subdiagram K i+1 contains K i and an additional square s such that at least two consecutive edges of s are contained in K i .Choosing a square s and adding it to K i to obtain K i+1 will be referred to as pushing a square.
Note that the sequence of diagrams K 0 , . . .K n is indeed finite, as Area(K i+1 ) = Area(K i ) + 1 for each i, and thus Area(D By construction, every dual curve h in D 0 starting in γ ′ must exit in γ.Indeed, every square S that is being pushed has at least two consecutive edges on γ (in the first step) or on some K i (in general).Thus, the 2 dual curves emanating from S either directly terminate on γ or enter K i , crossing some of the previously added squares.By induction on the area of K i , we can thus conclude that these dual curves terminate on γ.
Then for any weakly reduced disc diagram (D, ∂D) → (X * , X) with ∂D = γ 1 γ 2 where γ 1 , γ 2 are d p -geodesics and the subdiagram D ′ obtained from its sandwich decomposition The idea of the proof is as follows.Proceeding by contradiction, if D ′ is not a ladder, then D ′ contains a shell whose outerpath is disjoint from the endpoints q, q ′ of γ 1 , γ 2 .Using this shell and Lemma 4.6, we construct a path γ with endpoints q and q ′ and with shorter piece-length than γ 1 , which contradicts the fact that γ 1 is a piece geodesic (see Figure 6).
Proof.Suppose to the contrary that D ′ is not a ladder.We will derive a contradiction with the fact that γ 1 , γ 2 are d p -geodesics.By Lemma 2.7, D ′ has at least three exposed cells, i.e. shells of degree ≤ 4, corners and/or spurs.Two of those exposed cells might contain q and q ′ , but there still must be at least one other exposed cell whose boundary path is disjoint from both q and q ′ .By construction of D ′ in Construction 5.3, there are no corners or spurs contained in the interior of the paths γ 1 and γ 2 , so we conclude that there must be a shell S of degree ≤ 4 in D ′ with the outerpath Q contained in γ 1 or γ 2 .Up to switching names of γ 1 and γ 2 , we can assume that Q is contained in γ 1 .Let R denote the innerpath of S in D ′ .
Let e ℓ and e r be the leftmost (first) and the rightmost (last) edge of R, and let h ℓ , h r be their dual curves in D 1 .By Construction 5.2 h ℓ , h r exit D 1 in γ 1 .Let γ ′ 1 be the minimal subpath of γ 1 that contains the edges dual to h ℓ , h r .
Let H ℓ , H r be the hyperplanes of X extending h ℓ , h r respectively.Let ℓ, r be combinatorial paths in D 1 parallel to h ℓ , h r and starting at the two endpoints of the path Q, respectively.
Consider a minimal complexity square disc diagram E with boundary ∂E = ℓQrγ ′ 1 where ℓ and r are combinatorial paths contained in N (H ℓ ), N (H r ).In particular, ℓ and r do not intersect H ℓ and H r respectively.Such a diagram E exists since we can choose a subdiagram of D 1 .Amongst all possible choices of ℓ, r and E we pick a diagram with minimal area.A feature of the choice of E is that it has no cornsquares in the interiors of ℓ and r, as otherwise we could push that cornsquare out and reduce the area.Up to possibly replacing Q with another path with the same endpoints contained in the same cone, we can assume that Q has no cornsquares either.We will assume that this is the case for the remainder of the proof.
We will be applying Lemma 4.6 to E, so we first verify that the assumptions are satisfied.By Lemma 2.7, |Q| p ≥ p − 4 > 3. Next, we claim that every dual curve starting in ℓQr exits E in γ ′ 1 .The cases of dual curves starting in ℓ and r are analogous, so we only explain the argument for ℓ.Consider the subdiagram E ′ = E ∪ S of D ′ .Let e be an edge in ℓ.Note that the dual curve h to e in E ′ cannot terminate on Q, since this would imply that there is a cornsquare on Q.If h terminates on r, then h is parallel to Q, and therefore Q is a single wall-piece, contradicting the C(p) condition.Thus, h must terminate on γ ′ 1 .Let now e be an edge of Q, and h its dual curve in E. We already know that h cannot exit E in ℓ or r.If h exited E in Q, it would either yield a cornsquare in the interior of Q, contradicting the choice or Q, or it would yield a bigon formed from 2 squares glued along a pair of adjacent edges, contradicting the minimal complexity of E.
Since no dual curve in E crosses ℓQr twice, there are no cornsquares in none of ℓ, Q, and r, and | p which contradicts the fact that γ 1 is a piece-geodesic, completing the proof.□ We now combine the previous ingredients to finish the proof of Theorem 5.1.Here is an outline of the proof.We consider a piece-geodesic bigon and want to prove that it is thin in the piece-metric.We apply the reduction moves from Definition 2.1 to obtain a new bigon.We then apply Lemma 5.4 to show that the middle layer of its sandwich decomposition is a ladder, and thus it is thin by Proposition 4.5.Finally, we use Corollary 4.4 to deduce that the other layers are also thin, and consequently the original bigon is thin.
Proof of Theorem 5.1.We prove that the coned-off space ( X (0) , d p ) is δ ′ -hyperbolic for some δ ′ by showing that it satisfies the bigon criterion (Proposition 3.1).
We now describe the transformation from λ i to λi and from λ ′ i to λ′ i , for each reduction move.In each case the change will occur only within a subdiagram B that is transformed to B by the reduction.See Figure 7.We assume λ i intersects the interior of B. We note that it might happen that both λ 1 , λ 2 intersect B, in which case we apply the transformations to both λ 1 , λ ′ 1 and λ 2 , λ ′ 2 , according to the rules described below.It might happen that paths λ 1 , λ 2 overlap, but at no step they intersect transversally, i. i for i = 1, 2 satisfying the required conditions.We note that in the case where both λ 1 , λ 2 intersect B, the choices of λ1 , λ2 ensure that λ1 , λ2 do not intersect transversally.
Consider reduction move (4).Let B be a subdiagram associated to a cornsquare s and its dual curves ending on a cone-cell C. We set λi ∩ B to the combinatorial path with the same length and endpoints in B and maximizing the area of D Finally, consider reduction move (5).Let B be a subdiagram consisting of a square s overlapping with a cone-cell C along a single edge e.Thus λ i ∩ B = e.We set λi ∩ B to the path ∂s − e.We also set λ′ i ∩ B = λ ′ i ∩ B. See the last diagram in Figure 7.
Working under the assumption of weakly reduced: We now assume that (D, λ 1 , λ ′ 1 , λ 2 , λ ′ 2 ) satisfies conditions (a)-(c) above and that D is weakly reduced.Following the notation in (b), we claim that either D 1 ∪ D ′ ∪ D 2 is the sandwich decomposition of D, or we can push squares into D 1 and D 2 modifying λ i , λ ′ i while preserving conditions (a)-(c).Since λ i ∩ D □ = λ ′ i ∩ D □ and λ ′ i is a geodesic in X * , no square s in D ′ □ has three sides on λ i .So if a square s on λ i can be pushed into D i , then s must have two consecutive edges a, b on λ i .Let λ i = ℓ i abr i where ℓ i , r i are subpaths of λ i − ab.Likewise, let λ ′ i = ℓ ′ i abr ′ i .Finally, define λi = ℓ i cdr i and λ′ i = ℓ ′ i cdr ′ i where c, d are the other two edges of s.The quintuple (D, λ1 , λ′ 1 , λ2 , λ′ 2 ) satisfies conditions (a)-(c).Indeed, since |λ ′ i | = | λ′ i |, condition (c) is still satisfied.As this replacement does not affect D, nor the property of being (weakly) reduced, nor that D 1 and D 2 are square diagrams, conditions (a) and (b) are also preserved.We arrive at the sandwich decomposition after finitely many square-pushes.

Figure 1 .
Figure 1.Cone-cells in a disc diagram.In figures we will often omit the cell structure of cone-cells, unless needed.

Figure 2 .
Figure 2. The six reduction moves from Definition 2.1.
This is equivalent to D either being a single vertex or an edge, or containing a cut vertex.In particular, every degenerate disc diagram is singular.A square s is a cornsquare on a cone-cell C if a pair of dual curves emanating from consecutive edges a, b of c terminates on consecutive edges a ′ , b ′ of ∂D.Definition 2.1 (Reduction moves).We define six types of reduction moves.See Figure 2. (0) Cancelling a pair of squares s, s ′ meeting at one edge e in the disc diagram, whose map to X * factors through a reflection identifying them.That is, cutting out e ∪ Int(s) ∪ Int(s ′ ) and then glueing together the paths ∂s − e and ∂s ′ − e. (1) Replacing a minimal bigon-diagram, i.e. a disc subdiagram containing two dual curves intersecting each other twice, which is not contained in any other such subdiagram, with a lower complexity square disc diagram with the same boundary.(2) Replacing a pair of adjacent cone-cells with a single cone-cell.(3) Replacing a cone-cell with a square disc diagram with the same boundary.(4) Absorbing a cornsquare s to a cone-cell C, i.e. replace a minimal subdiagram containing C and the two dual curves starting at C and ending in s with a lower complexity disc diagram with the same boundary and containing a cone-cell C ∪ s ′ for some square s ′ .(5) Absorbing a square with a single edge in a cone-cell into the cone-cell.Definition 2.2 (Reduced and weakly reduced disc diagram).A disc diagram D → X * in a cubical presentation is • reduced if no moves (0) -(5) from Definition 2.1 can be performed in D.

Figure 3 .
Figure 3. Blue paths are contiguous pieces, and yellow paths are pieces but not contiguous pieces.

Figure 4 .
Figure 4. Example of a ladder.
Proposition 4.3.Let X * = ⟨X | Y 1 , . . ., Y s ⟩ be a cubical presentation satisfying the C(p) condition for p ≥ 2, and where X, Y 1 , . . ., Y s are compact.Then ( X (0) , d p ) is quasi-isometric to ( X (0) , d) where d is the standard combinatorial metric.Moreover, there is a uniform bound on the d p -diameters of cones.Proof.Indeed, d p (a, b) ≤ d(a, b) for all a, b ∈ X (0) , and by Lemma 2.5 there is an upper bound M on the combinatorial length of pieces, so we also have that d(a, b) ≤ M d p (a, b).

Construction 5. 2 (
Square pushes).Let D be a minimal complexity disc diagram, and let γρ = ∂D.Let λ be a path with the same endpoints as γ and lying in the cubical convex hull of γ, such that γλ bounds a disc subdiagram D 0 of D of maximal area.In particular, D 0 is a square disc diagram, and D ′ = D − D 0 is a disc diagram with ∂D ′ = λρ, which has no corners contained in the interior of the path λ.The diagram D 0 can be obtained via a finite sequence of square pushes, i.e. a sequence of subdiagrams

Figure 6 .
Figure 6.On the left, notation in the proof of Lemma 5.4; on the centre, possible overlapping paths in the proof of 5.1; on the right, the impossible transversal intersections described in the proof of 5.1.
e. at each step they yield a decomposition of the diagram D into D 1 ∪ D ′ ∪ D 2 .See Figure 6.Consider reduction move (1).Let B be a bigon-subdiagram of D, which is to be reduced.Since this reduction move involves only squares, we have λ i ∩ B = λ ′ i ∩ B. Note that λ i ∩ B cannot join the two corners of B since λ ′ i is a combinatorial geodesic.Thus λ i ∩ B must be a combinatorial geodesic crossing both dual curves associated to B. For each λ i ∩ B we set λi ∩ B = λ′ i ∩ B to the combinatorial geodesic with the same endpoints in B and maximizing the area of D i .See the first diagram in Figure 7. Thus the reduction move yields a new disc diagram D with new paths λi , λ′ i .If λ ′ i coincides with λ i in B we set λ′ i ∩ B = λi ∩ B. See the second diagram in Figure 7. Otherwise, we set λ′ i ∩ B = λ ′ i ∩ B. See the third diagram in Figure 7. Again, in the case where both λ 1 , λ 2 intersect B, the choices of transformed paths ensure that no transversal intersection occurs.