Non‐planarity of SL(2,Z)$\operatorname{SL}(2,\mathbb {Z})$ ‐orbits of origamis in H(2)$\mathcal {H}(2)$

We consider the SL(2,Z)$\operatorname{SL}(2,\mathbb {Z})$ ‐orbits of primitive n$n$ ‐squared origamis in the stratum H(2)$\mathcal {H}(2)$ . In particular, we consider the 4‐valent graphs obtained from the action of SL(2,Z)$\operatorname{SL}(2,\mathbb {Z})$ with respect to a generating set of size two. We prove that, apart from the orbit for n=3$n = 3$ and one of the orbits for n=5$n = 5$ , all of the obtained graphs are non‐planar. Specifically, in each of the graphs we exhibit a K3,3$K_{3,3}$ minor, where K3,3$K_{3,3}$ is the complete bipartite graph on two sets of three vertices.

where the stratum ( 1 , … ,   ) is the subset of  g consisting of those translation surfaces (, ) where  has  zeros of orders  1 , … ,   .Each stratum is a complex orbifold of dimension 2g +  − 1, and in a natural coordinate system on these strata origamis can be thought of as the integer lattice points.This point of view was utilized by Eskin-Okounkov [3] and Zorich [14] to calculate the volumes of strata with respect to a natural measure.For more details on translation surfaces and their applications, we direct the reader to the surveys of Forni and the second author [4], Yoccoz [12], and Zorich [15].
Each stratum of translation surfaces admits a natural action of SL(2, ℝ) which restricts to an action of SL(2, ℤ) on the origamis in that stratum.In this paper, we consider the SL(2, ℤ)-orbits of primitive -squared origamis in the stratum  (2).These orbits were classified in the works of McMullen [9] and Hubert-Lelièvre [6].We note that an origami in (2) requires at least 3 squares.For  = 3 or  ⩾ 4 even there is a single SL(2, ℤ)-orbit of primitive -squared origamis in (2).For  ⩾ 5 odd, there are two SL(2, ℤ)-orbits of primitive -squared origamis in (2).These two orbits, defined more explicitly in Subsection 2.2, are called the A-and B-orbits, respectively.By considering SL(2, ℤ) = ⟨, ⟩, where each SL(2, ℤ)-orbit can be realized as a finite 4-valent graph which we will denote by   for  = 3 or  ⩾ 4 even, and for  ⩾ 5 odd by    , or    for the A-and B-orbits, respectively.It is a conjecture of McMullen that this family of graphs forms a family of expanders.Here we prove that, apart from  3 and   5 , these graphs are non-planar.That is, our main result is the following.
Theorem 1.1.The graphs   ,    , and    are all non-planar with the exception of  3 and   5 .
This provides indirect evidence for McMullen's conjecture.Indeed, it follows from the separator theorem for planar graphs of Lipton-Tarjan [8] that planar graphs cannot form an expander family.We direct the reader to the exposition of this fact in de Courcy-Ireland's recent work [2,Section 12] demonstrating that, for certain primes , Markoff graphs modulo  -another conjectured expander family -are non-planar.
Our proof relies on the theorem of Wagner [11] and Kuratowski [7] characterizing planar graphs in terms of forbidden minors.Recall that a graph  is realized as a minor inside a graph  if one can perform a sequence of edge-contractions, edge-deletions, and deletions of isolated vertices in order to transform the graph  into the graph .Wagner and Kuratowski's characterization can then be worded as follows: a graph  is planar if and only if it does not contain a  5 or  3,3 minor, where  5 is the complete graph on five vertices and  3,3 is the complete bipartite graph on two sets of three vertices.We prove the non-planarity claim of Theorem 1.1 by realizing a  3,3 minor in each case.
The planar graphs  3 and   5 are shown in Figures 3.1 and 3.2 at the end of Section 3.

Weierstrass curves and an alternate generating set
and forms the associated orbit graphs in this setting, which we shall denote by   ,    , and    , then computational experiments lead us to make the following conjecture.In other words, we conjecture that planarity in this setting exactly agrees with the range for genus zero components of   2 given by Mukamel.The planarity for  ⩽ 7 and of   9 is computationally confirmed and so the conjecture is that the remainder are all non-planar.The non-planarity of   has been confirmed for even  in the range 8 ⩽  ⩽ 16, of    in the range 9 ⩽  ⩽ 17, and of    in the range 11 ⩽  ⩽ 23.Finally, we also remark that there exists a generalization of the separator theorem of Lipton-Tarjan given by Gilbert-Hutchinson-Tarjan [5] from which it follows that a family of expanders must have genus tending to infinity.So, for us, if the family of graphs   , or   , are a family of expanders then the genus of these graphs must tend to infinity.In the Weierstrass curve setting, Mukamel [10, Corollary 1.3], building on work of Bainbridge [1], has shown that the genus of any component of   tends to infinity as  tends to infinity.

ORIGAMI PRELIMINARIES
Here we give an introduction to origamis and the classification of their SL(2, ℤ)-orbits in the stratum (2).In particular, we will discuss their algebraic description using pairs of permutations which we will use in the remainder of the paper.

Origamis
An origami is an orientable connected surface obtained from a collection of unit squares in ℝ 2 by identifying by translation left-hand sides with right-hand sides and top sides with bottom sides.See for example the surface in Figure 2.1 where sides with the same label are identified by translation.
An -squared origami can also be described by two permutations ℎ and  in the symmetric group Sym().These permutations are obtained as follows.First, we number the squares from 1 to .We then define ℎ to be the element of Sym() such that ℎ() =  if and only if the righthand side of the square labeled by  is identified with the left-hand side of the square labeled by .Similarly, we define  to be the permutation satisfying () =  if and only if the top side of the square labeled by  is glued to the bottom side of the square labeled by .For example, the origami in Figure 2.1 can be described by the pair (ℎ, ) = ((1, 2, 3), (1,3)).Since a different labeling of the squares could produce a different pair of permutations, an origami actually corresponds to a pair (ℎ, ) considered up to simultaneous conjugation of ℎ and .However, we will abuse notation in this article and denote an origami simply by the pair (ℎ, ).When we need to make it clear, we will use the notation (ℎ, ) ≃ (ℎ ′ ,  ′ ) if the pairs are equivalent by simultaneous conjugation.
We will also use the monodromy group () of an origami  which is defined to be the subgroup of Sym() generated by ℎ and ; that is, ((ℎ, )) = ⟨ℎ, ⟩.An origami is then said to be primitive if its monodromy group is a primitive subgroup of Sym().Topologically, an origami is said to be primitive if it is not a proper cover of another origami (other than the unit-square torus of which all origamis are a cover).Both these notions of primitivity are equivalent to one another.For more on the translation between the algebraic and topological descriptions of origamis we direct the reader to the thesis of Zmiaikou [13].
We remark that the definitions of the monodromy group and of the stratum of an origami are well defined in the sense that they are invariant under simultaneous conjugation of ℎ and .

Orbit classification
We now discuss the action of SL(2, ℤ) on origamis and the classification of orbits in the stratum (2).] .
The matrix  acts by horizontally shearing the origami to the right while the matrix  acts by vertically shearing the origami upward.
In particular, it follows that the number of squares, the stratum, the primitivity, and the monodromy group of an origami are invariant under the action of SL(2, ℤ).As such, it makes sense to discuss the SL(2, ℤ)-orbits of primitive -squared origamis in a given stratum.

Theorem 2.1 (McMullen
Let  ⩾ 5 be odd.Then there are two SL(2, ℤ)-orbits, the A-orbit and the B-orbit, of primitive squared origamis in (2).Every origami in the A-orbit has monodromy group the full symmetric group Sym(), while every origami in the B-orbit has monodromy group the alternating group Alt().

CONSTRUCTION OF THE MINOR
In this section, we construct the  3,3 minors in each of the orbits.

Even squared orbits
Let  ⩾ 4 be even and define the origamis  1 , … ,  6 as follows: By calculating [ℎ, ] in each of the above cases, it is easy to check that these origamis lie in (2).
We then have the following.
Proof.Let  denote the path in the  3,3 minor between   and   .We claim that the following paths realize the minor: We need only check that these paths do connect the claimed origamis and that they are pairwise disjoint.
It is readily checked that we have where Finally, we must check the paths 24 and 26.We observe that the origami  4 has a -orbit of length  which begins with  4 ,  3 ,  5 , and  2 .The remaining  − 4 points of the orbit form the path 24 and, since the conjugacy class of the first permutation remains unchanged along this orbit, we see that this path is disjoint from all of the paths considered so far.
For 26, we observe that Observe that for  = 4 we have  −1 ( 2 ) =  6 .We will assume  ⩾ 6 in the remainder.It can then be checked that  −1 ( 2 ) lies in the same -orbit as  6 .However,  ′′ =  −1 ( 6 ) =  2 • −1 ( 2 ) and so, to avoid  ′′ , we must complete the path 26 using powers of  −1 .The -orbit of  6 is of size  − 1 and so we require  −(−4) to take  −1 ( 2 ) to  6 .As  ′′ is the only origami in the -orbit of  6 that we have used already (apart from  6 , of course) we see that this path is disjoint from all of those we have considered above.This completes the proof.□

A-orbits
Let  ⩾ 5 be odd and define the origamis  1 , … ,  6 as follows: By calculating [ℎ, ] in each of the above cases, it is easy to check that these origamis lie in (2).Furthermore, they all have their monodromy group being the symmetric group and so they lie in the A-orbits.We then have the following.

B-orbits
Let  ⩾ 7 be odd and define the origamis  1 , … ,  6 as follows: Again, it is easy to check that these origamis lie in (2).Furthermore, they all have their monodromy group being the alternating group and so they lie in the B-orbits.We then have the following.Proof.Let  denote the path in the  3,3 minor between   and   .We claim that the following paths realize the minor: It can be checked that  −1 ( 2 ) and • −1 ( 5 ) lie in the same -orbit.However,  ′ =  −1 •• −1 ( 5 ) =  2 • −1 ( 2 ).As such, we must complete the path using powers of  −1 .The orbit of  −1 ( 2 ) is of length  and so we require  −(−3) to take  −1 ( 2 ) to • −1 ( 5 ).It can also be checked that no more of the origamis we have already considered are contained in this -orbit and so this path is disjoint from those we have constructed above, and so we are done.□ The graphs in Figures 3.1 and 3.2 demonstrating the planarity of  3 and   5 complete the work of this section and the proof of Theorem 1.1.Note that we have used directed edges in the figures so that the reader can more easily check the validity of the graphs.

FURTHER QUESTIONS
We finish with two natural questions.First: Question 4.1.Can one find generalizations of these structures that exist in the SL(2, ℤ)-orbits of primitive origamis in different strata?
We remark that the classification of these SL(2, ℤ)-orbits is open in general.Second, the work of de Courcy-Ireland for Markoff graphs modulo  gave, for certain primes , a construction of a  3,3 minor whose path lengths did not depend on the prime .They called such a construction a 'local' construction.In our case, this would correspond to finding a  3,3 minor in   whose paths have lengths that do not depend on .As such, we ask the following.

Conjecture 1 . 2 .
The graphs   ,    , and    are planar for  ⩽ 7. The graph   9 is planar.All remaining graphs are non-planar.

F I G U R E 2 . 2
The action of  on the permutations ℎ and  of an origami.The group SL(2, ℤ) acts on origamis by its natural action on the plane.Indeed, up to cutting and pasting, unit squares are mapped to unit squares and parallel sides are sent to parallel sides.Consider SL(2, ℤ) = ⟨, ⟩, where

F I G U R E 3 . 2
The graph   5 is the undirected version of this graph.

Question 4 . 2 .
Does there exist a 'local' construction of a  3,3 (or  5 ) minor in the SL(2, ℤ)-orbits of primitive origamis in (2)?A C K N O W L E D G E M E N T SThe first author is thankful for support from the Heilbronn Institute for Mathematical Research.
[9]eichmüller curve is an algebraic and isometric immersion of a finite-volume hyperbolic Riemann surface into the moduli space  g of Riemann surfaces of genus g.McMullen[9]classified all of the Teichmüller curves in genus two.The main source of such Teichmüller curves are the so-called Weierstrass curves   parameterized by integers  ⩾ 5 with  ≡ 0 or 1 modulo 4.These curves consist of those Riemann surfaces  ∈  2 whose Jacobians admit real multiplication by the quadratic order   ∶= ℤ[]∕⟨ 2 +  + ⟩, ,  ∈ ℤ with  =  2 − , and for which there exists a holomorphic one-form  on  such that (, ) ∈ (2) and   ⋅  ⊂ ℂ ⋅ .A translation surface (, ) ∈ (2) projects to   2 if and only if (, ) is an -squared origami.Corollary 1.4] that all of the components of   2 have genus zero for  ⩽ 7 and one of the components has genus zero for  = 9.We see that this planarity range does not agree with the planarity of our graphs   .However, if one considers the generating set {, } for SL(2, ℤ) where