Counting lattices in products of trees

A BMW group of degree $(m,n)$ is a group that acts simply transitively on vertices of the product of two regular trees of degrees $m$ and $n$. We show that the number of commensurability classes of BMW groups of degree $(m,n)$ is bounded between $(mn)^{\alpha mn}$ and $(mn)^{\beta mn}$ for some $0<\alpha<\beta$. In fact, we show that the same bounds hold for virtually simple BMW groups. We introduce a random model for BMW groups of degree $(m,n)$ and show that asymptotically almost surely a random BMW group in this model is irreducible and hereditarily just-infinite.


Introduction
Given n ∈ N, let T n denote the regular tree of valence n.A BMW group of degree (m, n) is a subgroup of Aut(T m ) × Aut(T n ) that acts simply transitively on the vertex set of T m ×T n .Using these groups, Wise [Wis07] and Burger-Mozes [BM00b] produced the first examples of non-residually finite and virtually simple CAT(0) groups respectively.BMW groups have been extensively studied and have rich connections to the study of automata groups and commensurators (see Caprace's survey [Cap19]).
In this paper, our goal is two-fold: (1) estimate the number of BMW groups and virtually simple BMW groups up to abstract commensurability, and (2) define and study a random model for BMW groups.Although conceptually related, the two parts are independently presented.
Counting BMW groups.Let BMW(m, n) be the set of all BMW groups of degree (m, n) up to conjugacy in Aut(T m ) × Aut(T n ).Let ≃ be the equivalence relation of abstract commensurability, i.e., the groups Γ and Λ satisfy Γ ≃Λ if they have isomorphic finite index subgroups.In analogy to counting results for hyperbolic manifolds (see Remark 1.1), Caprace [Cap19,Problem 4.26] asks for an estimate on the number of abstract commensurability classes of BMW groups of degree (m, n) as m, n → ∞.Addressing this question, we give the following result: Theorem A. There exist 0 < α < β such that, for all sufficiently large m and n, (mn) αmn ≤ BMW vs (m, n) ≃ ≤ BMW(m, n) ≃ ≤ (mn) βmn .
where BMW vs (m, n) is the collection of BMW groups, up to conjugacy, of degree (m, n) that contain an index 4 simple subgroup.
All BMW groups contain an index 4 normal subgroup, so the index in the above theorem is as small as possible.
Remark 1.1.Compare the above result with the bounds obtained by [BGLM02,GL14]: there exist 0 < α ′ < β ′ such that the number of commensurability classes of hyperbolic manifolds of volume at most v is bounded between v α ′ v and v β ′ v .
A random model for irreducible BMW groups.The random model we define is based on a combinatorial description of BMW groups (more precisely, of involutive BMW groups) given in §2.1.We postpone the definition of the model to §5, and only highlight its main properties in Theorem B below.This model does not capture all possible BMW groups.It was chosen predominantly for its relative ease of computations on the one hand, and its naturality on the other.A BMW group is irreducible if it does not contain a subgroup of finite index that is isomorphic to the direct product of two free groups.A group is just-infinite if it is infinite and has only finite proper quotients.It is hereditarily just-infinite if all its finite-index subgroups are just-infinite.

Theorem B.
A random BMW involution group of degree (m, n), with n > m 5 , is hereditarily just-infinite and, in particular, is irreducible with probability at least 1 − C m , where C is a constant that is independent of m and n.Remark 1.2.In fact, when n > m 5 we have that: • An arbitrary BMW group of degree (m, n) is irreducible if and only if it has non-discrete projections to both its factors.Something stronger happens in the random model: asymptotically almost surely, the projection to Aut(T m ) (resp. to Aut(T n )) of a random BMW group in the model contains the universal groups U (A m ) (resp.U (A n )).• By the previous remark and a rigidity theorem for BMW groups [BMZ09, Theorem 1.4.1](see also Theorem 3.8), asymptotically almost surely, two random BMW groups are not isomorphic.
We give the following conjecture regarding this random model: Conjecture 1.3.In the above range of m and n, a random BMW group is asymptotically almost surely not residually finite and consequently is virtually simple by Theorem B.
More generally, we conjecture the following: Conjecture 1.4.As m, n → ∞, the proportion of virtually simple BMW groups of degree (m, n), up to conjugacy, tends to 1.
Positive evidence for Conjecture 1.4 has been given by Rattagi [Rat04] and Radu [Rad20] for small values of m and n.
Outline.In §2 we discuss involutive BMW groups and a combinatorial description of them.In §3, we bound the number of BMWs from above, and in §4, we prove the more difficult lower bound, giving Theorem A. In §5 we present our random model for BMW groups.Next, in §6 and §7, we show that the local actions are alternating or symmetric with high probability.Finally, in §8 we prove Theorem B. We note that sections §3 and §4 can be read independently of §5, §6, §7 and §8 (and vice versa).

Involutive BMW Groups
An involutive BMW group Γ of degree (m, n) is a BMW group such that for every edge e of T m × T n , there is some g ∈ Γ that interchanges the endpoints of e.Since Γ acts simply transitively on vertices, such an element g must be an involution.We let BMW inv (m, n) denote the set of all involutive BMW groups of degree (m, n), up to conjugacy in Aut(T m ) × Aut(T n ).A BMW group of degree (m, n) is irreducible if and only if the projection of Γ to either Aut(T n ) or Aut(T m ) is not discrete [BM00b, Proposition 1.2].
Any tree T is bipartite; let Aut + (T ) be the index 2 subgroup of Aut(T ) preserving the bi-partition of T .Given a BMW group Γ of degree (m, n), denote by the index 4 subgroup of Γ preserving the bipartitions of T m and T n .This subgroup is always torsion-free [Rad20, Lemma 3.1].
2.1.Structure sets.In this section we describe structure sets which encode presentations for BMW groups.For the rest of this article, we fix countable indexing sets a 1 , a 2 , . . .and b 1 , b 2 , . . ., and for each k ∈ N, we set A k ∶= {a 1 , . . ., a k } and for every a ∈ A m and b ∈ B n , {a, b} is a subset of exactly one set in S. Let S m,n denote the set of all (m, n)-structure sets.
For a structure set S, denote by R S the set of words in A m ⊔ B n defined as Remark 2.2.In the definition of a structure set, the elements a i , a j (and similarly b k , b l ) of a set {a i , b k , a j , b l } ∈ S are not assumed to be distinct, so some {a i , b k , a j , b l } ∈ S may have fewer than 4 elements.We often still write repeating elements in these subsets -e.g.
Remark 2.3.A useful point of view on structure sets is given by partitions of the complete bi-partite graph: Let K m,n be the complete bi-partite graph on A (combinatorial) square complex is a 2-complex in which 2-cells (squares) are attached along combinatorial paths of length four.A VH-complex is a square complex in which the set of 1-cells (edges) is partitioned into vertical and horizontal edges such that the attaching map of each square alternates between them.We regard the product T m × T n of trees as a VH-complex where an edge is horizontal if it lies in T m × {v} for some v ∈ T n , and vertical if it lies in {v} × T n for some v ∈ T m .

Definition 2.4 (Marking)
. A marking M on T m × T n is a choice of a base vertex o ∈ T m × T n and an identification of the horizontal (resp.vertical) edges incident to o with A m (resp.B n ).An element g ∈ Aut(T m ) × Aut(T n ) is said to fix M if g fixes o and fixes all edges adjacent to o.
Fix a marking M on T m ×T n with base vertex o.Let BMW M (m, n) be the set of all involutive BMW groups of degree (m, n), up to an automorphism fixing M. In other words, two BMW groups Γ and Γ ′ of degree (m, n) are equal in BMW M (m, n) if and only if Γ = gΓ ′ g −1 where g is an element of Aut(T m ) × Aut(T n ) that fixes M. Let S m,n be the set of all (m, n)-structure sets.We now describe how to obtain a bijection: Let Γ be an involutive BMW group of degree (m, n).As Γ is involutive, each edge of T m ×T n is stabilized by a unique element of Γ.By a slight abuse of notation, we let a i (resp.b i ) denote the element of Γ that stabilizes the edge that is adjacent to o with label a i (resp.b i ).As Γ acts freely and transitively on the vertices of T m × T n , its action induces a well-defined, Γ-invariant labeling of the edges of T m × T n which we generally call the Γ-induced labeling.Moreover, it is readily checked that the 1-skeleton of T m × T n with this labeling is the Cayley graph for Γ with generating set {a 1 , . . ., a m , b 1 , . . ., b n } (where bigons in this Cayley graph are collapsed to edges).
We now describe how to form an (m, n)-structure set S associated to Γ. Let S be the collection of subsets {a i , b k , a j , b l } such that there exists a square in T m × T n whose edges are labeled a i , b k , a j , b l with respect to the Γ-induced labeling.Note that since Γ acts simply transitively on the vertices of T m × T n and preserves the Γ-induced labeling, it suffices to only consider the squares containing o.
To show that S is a structure set, let a ∈ A m and b ∈ B n .There exists a unique square s which contains both edges incident to o labeled by a and b.Since Γ acts simply transitively on vertices, any other square which contains two edges labeled by a and b is in the orbit of s and consequently its edges have the same labels as s.Thus, there is a unique {a i , b k , a j , b l } ∈ S containing a and b.So, S is indeed an (m, n)-structure set.We say that S is the structure set associated with Γ, and we define Φ([Γ]) = S, where [Γ] is the equivalence class in BMW M (m, n) containing Γ.
Additionally, we conclude that Γ has the presentation This follows since we can take the 1-skeleton of T m ×T n , label it with the Γ-induced labeling (i.e., form the Cayley graph for Γ) and attach 2-cells corresponding to the relations R S .The resulting complex can also be obtained from the Cayley complex for Γ by collapsing each bigon corresponding to the relations a 2 and b 2 to an edge.In fact, it is just T m × T n with the Γ-induced edge labeling.We now need to check that Φ is well defined.Suppose that Γ ′ is an involutive BMW group of degree (m, n) that is conjugate to Γ by some g ∈ Aut(T m )×Aut(T n ) that fixes M.Then, as g fixes M, the Γ-induced labeling and Γ ′ -induced labeling of T m × T n agree on all squares that contain o.It follows by construction that the structure sets associated to Γ and Γ ′ are equal.Consequently, Φ is well-defined.
We now check that Φ is injective.Suppose that Γ and Γ ′ are involutive BMW groups of degree (m, n) and that Φ([Γ]) = Φ([Γ ′ ]).Consequently, the Γ-induced labeling and the Γ ′ -induced labeling of T m × T n agree on all squares which contain o.It now readily follows that Γ ′ is conjugate to Γ by some element g ∈ Aut(T m ) × Aut(T n ) that fixes the marking M. Thus, Γ and Γ ′ are equal in BMW M (m, n).
Finally, we check that Φ is surjective.Let S be an (m, n)-structure set.Then the group Γ with presentation as in (2.1) is an involutive BMW group (see for instance [Cap19]).Moreover, Φ([Γ]) = S.We have thus shown: Proposition 2.5.Let M be a marking of T m × T n .There is a bijection Moreover, for each S ∈ S m,n , each representative of Φ −1 (S) has the presentation Let g ∈ Aut(T m ) × Aut(T n ) be an automorphism.We describe how g acts on markings.Let M be a marking of T m × T n with base vertex o.Then g induces a new marking M ′ = gM whose base vertex is o ′ = go and such that the label of an edge e adjacent to o ′ is equal to the label of g −1 e under the marking M.This action of g also induces a bijection Let S be an (m, n) structure set, and let µ ∈ Sym(A m ) and ν ∈ Sym(B m ) be permutations.We can form a new (m, n)-structure set S ′ by applying the permutation We say that S ′ is a relabeling of P induced by µ and ν.
Let Γ ′ = gΓg −1 for some g ∈ Aut(T m ) × Aut(T n ).Since Γ acts vertex transitively, we may assume, without loss of generality, that g fixes o.Thus, g induces permutations µ ∈ Sym(A m ) and ν ∈ Sym(B m ) on the labels (in the marking M) of the edges incident to o.It readily follows that the structure set of S ′ of Γ ′ is obtained from the structure set S of Γ by relabeling.
Thus, if Γ and Γ ′ are conjugate BMW groups, then their associated structure sets are the same up to a relabeling, regardless of a choice of marking.Conversely, suppose that S ′ is an (m, n)-structure set that is a relabeling of a structure set S induced by µ ∈ Sym(A m ) and ν ∈ Sym(B n ).Then we can choose a marking M on T m × T n and a BMW group Γ whose associated structure set is P .Additionally, we can choose a g ∈ Aut M (T m × T n ) so that the induced action of g on the labels of the horizontal and vertical edges around o is given by µ and ν respectively.From this, we have that gΓg −1 is a BMW group conjugate to Γ whose structure set is S ′ .We have thus shown the following: Proposition 2.6.There is a bijection

Local actions.
Let X be a locally finite graph.For every vertex v ∈ V (X), let E(v) be the set of edges of X incident to v. If a group Γ acts on X, the local action of Γ on X at the vertex v is the induced action of Stab Γ (v) on the set E(v).By abuse of terminology, we will also refer to the image of the action Stab Γ (v) → Sym(E(v)) ≅ Sym(n) as the local action, where n = E(v) .The local actions of Γ ≤ Aut(T m ) × Aut(T n ) are the local actions of Γ on T m and T n .More specifically, we call the local action of Γ on T m the A-tree local action, and the local action of Γ on T n the B-tree local action.
We show how to read off the local action of an involutive BMW group of degree (m, n) from the corresponding structure set.First note that since a BMW group of degree (m, n) acts transitively on the vertices of T m × T n , its local actions on T m (resp.T n ) at different vertices are conjugate actions.We can thus refer to the  Theorem 2.8 (Burger-Mozes).Let m, n ≥ 6, let Γ be an irreducible BMW group of degree (m, n), and assume that the local actions of Γ on T m and T n contain the alternating groups Alt(m) and Alt(n) respectively.Then Γ is hereditarily justinfinite.
For a group Γ, its finite residual Γ (∞) is the intersection of all finite-index subgroups of Γ.The following widely known lemma is a tool to prove virtual simplicity of a group.Lemma 2.9.If a group Γ is hereditarily just-infinite and is not residually finite, then Γ (∞) is a finite index, simple subgroup of Γ.
Proof.Since Γ is not residually finite, the finite residual Γ (∞) is a non-trivial normal subgroup of Γ.By assumption Γ is just-infinite, thus Γ (∞) must have finite index in Γ.As Γ is hereditary just infinite, Γ (∞) is itself just-infinite, and thus cannot contain any non-trivial infinite-index normal subgroup.On the other hand, by definition and since Γ (∞) has finite index in Γ, Γ (∞) cannot have any non-trivial finite-index normal subgroup.Thus, Γ (∞) is simple.

Upper bounds on BMW counts
In this section we give upper bounds for the number of conjugacy classes of involutive BMW groups.We then use a result of Burger-Mozes-Zimmer to bound the number of BMW groups that are abstractly commensurable to a given BMW group with primitive local actions and simple type-preserving subgroup.

Upper bound on conjugacy classes of involutive BMWs.
Proposition 3.1.There are at most (mn) mn conjugacy classes of involutive BMW groups of degree (m, n).
This function is well-defined by the definition of a structure set.We can reconstruct S from f S by

Thus we get an injective map
where the equality BMW M (m, n) = S m,n follows from the bijection in Proposition 2.5.Remark 3.2.For general (not necessarily involutive) BMW groups, using the (m, n)-datum defined in [Rad20] (which is analogous to structure sets defined here), a similar proof gives a number β > 0 such that the number of conjugacy classes of BMW groups is bounded by (mn) βmn .

(m, n)-complexes and type-preserving subgroups.
In this subsection we associate to each involutive BMW-group a certain edge-labeled square complex that completely describes the group.These complexes will allow us to deduce a count on the number of involutive BMW groups with the same type-preserving subgroup up to conjugation.
(2) There are exactly m edges between v 00 and v 10 (resp.v 01 and v 11 ), all of which are horizontal and labelled by distinct elements of A m .(3) There are exactly n edges between v 00 and v 01 (resp.v 10 and v 11 ), all of which are vertical and labelled by distinct elements of B n .(4) For each horizontal edge e 1 and vertical edge e 2 of Y , there is a unique square containing both e 1 and e 2 .(5) There is a label-preserving vertex-transitive action on Y .Remark 3.4.By (4) above, an (m, n)-complex contains exactly mn squares.We also note that the label preserving automorphism group of an (m, n)-complex is precisely Z 2 × Z 2 .Lemma 3.5.Fix a marking on T m × T n , and let Γ be an involutive BMW group of degree (m, n).Label the edges of Conversely, let Y be an (m, n)-complex.Then the set S, consisting of the subsets Proof.Let Γ be as in the statement of the lemma.The type-preserving subgroup Γ + < Γ acts freely, so we can consider the quotient complex Z ∶= Γ + (T m × T n ).As edge labels pass to the quotient and as Γ Γ + ≃ Z 2 × Z 2 acts transitively on the vertices of Z, Z is an (m, n)-complex as required.The proof of the converse statement follows from Definitions 2.1 and 3.3.
We will need the following lemma counting the number of possible (m, n)complexes with isomorphic square complexes.Lemma 3.6.Given any (m, n)-complex C, there are at most 2(n!m!) 2 distinct (m, n)-complexes that are isomorphic to C as unlabeled square complexes (i.e.isomorphic via a cellular isomorphism that does not necessarily preserve labels or the VH-structure).
Proof.Let Y be an the underlying square complex of an (m, n)-complex C.There are at most two ways of giving Y a suitable V H-structure.After choosing a V Hstructure, Y has 4 vertices, 2m horizontal edges and 2n vertical edges, and there are at most (n!m!) 2 ways to choose labels to obtain an (m, n)-complex.Thus there are at most 2(n!m!) 2 possible (m, n)-complexes that are isomorphic to Y as unlabeled square complexes.
The next lemma bounds the number of involutive BMW groups with conjugate type-preserving subgroups.
as unlabeled square complexes.By Lemma 3.6, Y 2 can be one of at most 2(n!m!) 2 possible (m, n)-complexes.By Lemma 3.5 and Proposition 2.5, such an (m, n)-complex completely determines gΛg −1 up to a conjugation.The lemma now follows.
Recall that a subgroup F ≤ Sym(n) is primitive if no non-trivial partitions of {1, . . ., n} is stabilised by F .The following theorem is a reformulation of a result of Burger-Mozes-Zimmer, which builds on a superrigidity theorem of Monod-Shalom [MS04]: Recall that two groups are abstractly commensurable if they have isomorphic finite index subgroups.The previous theorem implies the following: Proposition 3.9.Let Γ be an involutive BMW group of degree (m, n) with primitive local actions and with Γ + simple.Then, up to conjugacy, there are at most 2(n!m!) 2 involutive BMW groups of degree (m, n) that are abstractly commensurable with Γ.
Proof.Let Λ be an involutive BMW group of degree (m, n) that is abstractly commensurable to Γ.As Γ + is simple, Λ contains a finite index subgroup H that is isomorphic to Γ + .It follows from Theorem 3.8 that H is type-preserving and has four orbits of vertices, so H = Λ + .By Theorem 3.8, Γ + is conjugate to Λ + .The result now follows from Lemma 3.7.

Counting commensurability classes of BMW groups
In order to count commensurability classes of BMW groups, we first define a partial structure set S 0 (see definition below).We show in Theorem 4.2 that if Γ is an involutive BMW group whose structure set contains S 0 , then the index 4 subgroup Γ + is simple.We then deduce the lower bound of Theorem A by showing that there are sufficiently many such Γ.
such that for every a ∈ A m and b ∈ B n at most one subset of S 0 contains both a and b.
It is straightforward to check that for any 1 ≤ i ≤ m and 1 ≤ k ≤ n, {a i , b k } is a subset of at most one set in S 0 , making S 0 a partial structure set.Theorem 4.2.If an (m, n)-structure set S contains S 0 , then its associated involutive BMW group Γ satisfies that Γ + is simple.
To prove the above theorem, we need to show a few lemmas first.Throughout, let S be an (m, n)-structure set containing S 0 and let Γ be its associated involutive BMW group.Proof.We show the claim for the B-tree local action.The argument is similar for the A-tree local action.Let α 1 , . . ., α m be the B-tree local involutions of Γ.By Lemma 2.7 and as S ∆ , S AR ⊂ S, we get that for 1 , where γ 4 is some unknown permutation in Sym( 6, n ).Similarly, as S BR , S BC , S AB , S A ⊂ S we get that for 1 ≤ i ≤ 3, α 4+i = id ×α ′ i where id is the identity permutation of Sym( 1, 5  Proof.Clearly Γ (∞) ≤ Γ + , thus it suffices to show that Γ (∞) has index 4 in Γ.More precisely, we show that Γ Γ (∞) ≃ Z 2 × Z 2.
In claim 2 below, we prove that bi = b for all 1 ≤ i ≤ n and āi = ā for all 1 ≤ i ≤ m.The lemma follows from this as Γ Γ (∞) is generated by ā, b and satisfies the relations We first prove the following claim, showing that certain generators of Γ are equal in the quotient.
Claim 1: āi = āj for all i, j ≥ 5, and bk = bl for all k, l ≥ 6 We show that bk = bl for all k, l ≥ 6.The second claim follows from a similar argument.
Let (k, l) be an edge of G.We have that α as generators are involutions).Since G is not bipartite and is connected, any two vertices of G are connected by an even length path.It follows that for any k and l such that 6 ≤ k < l ≤ n, there is an even number p such that bl = āp bk āp .As ā2 = 1, we deduce that bk = bl .The claim follows.
Claim 2: ā = āi for all 1 ≤ i ≤ m , and b = bk for all 1 ≤ k ≤ n By Claim 1, we can define ā′ ∶= āi for all i ≥ 5 and b′ ∶= bk for all k ≥ 6.To prove Claim 2, we need to show that ā = ā′ and b = b′ .We show ā = ā′ .The second statement follows from a similar argument.
First note that since S AR ⊂ S the word In the quotient, this relation becomes āb ′ āb ′ = 1 which implies that ā commutes with b′ .Next, since S M ⊂ S the word a 4 b n a m b n−1 ∈ R S is a relation in Γ.In the quotient, this gives āb ′ ā′ b′ = 1.As b′ commutes with ā, we see that ā = ā′ .The claim follows.
We are now ready to prove Theorem 4.2.
Proof of Theorem 4.2.By Lemma 4.3, the local actions of the group Γ are the full symmetric groups on m and n elements.Moreover, Γ is irreducible and not residually finite as it contains ∆ (see [Cap19, Proposition 4.2 vii)]).The theorem then follows from Theorem 2.8, Lemma 2.9 and Lemma 4.4.

Lemma 4.5.
There exists a number α > 0 such that, for all integers m > 0 and n > 0, the number of (m, n)-structure sets is at least (mn) αmn .
Proof.Without loss of generality assume that m ≤ n.Let I n ⊆ Sym(n) be the subset of all involutions.For any m involutions, α 1 , . . ., α m ∈ I n we can define a structure set Therefore, there are at least (I n ) m different (m, n)-structure sets.By [CHM51, Theorem 8], the number of involutions in Sym(n) is for large n.Thus the number of structure sets of degree (m, n) is at least mn ≥ (mn) where the last inequality follows from m ≤ n.By choosing α small enough, we get the claim for all m and n (not just large enough n).
Theorem 4.6.There exists a number α > 0 such that, for all sufficiently large natural numbers m and n, there are at least (mn) αmn pairwise non-commensurable, involutive BMW groups Γ of degree (m, n) such that Γ (∞) = Γ + is simple.
Proof.Note that the partial structure set S 0 has no element containing {a i , b k } where i ∈ 11, m − 3 and k ∈ 12, n − 3 .Set m ′ , n ′ to be the number of elements in those integer intervals respectively -namely m ′ = m − 13 and n ′ = n − 14.We see that we can add to S 0 any partial structure set supported on those elements.Using Lemma 4.5, there is some α ′ > 0 so that there are at least (m ′ n ′ ) α ′ m ′ n ′ different ways of extending S 0 to a partial structure set S ′ .By further adding the sets {a i , b j , a i , b j } to S ′ for any i, j such that {a i , b j } is not contained in an element of S ′ , one obtains a structure set S. Therefore there are at least (m ′ n ′ ) α ′ m ′ n ′ structure sets containing S 0 .Additionally, a BMW group Γ associated to such a structure set satisfies that Γ (∞) = Γ + is simple by Theorem 4.2.We now give a lower bound for the number of commensurability classes of involutive BMWs associated to structure sets containing S 0 .By Proposition 3.9 the commensurability class of such a BMW group has at most 2(m!n!) 2 structure sets.Therefore, by the previous paragraph, there are at least (m ′ n ′ ) α ′ m ′ n ′ 2(m!n!) 2 commensurability classes of such involutive BMW groups of degree (m, n).By using m! ≤ m m and n! ≤ n n and m ′ ≥ m 2 and n ′ ≥ n 2 one gets the desired lower bound.

A Random Model for BMW groups
For each even n ∈ N, let F n ⊆ Sym(n) be the subset of fixed-point-free involutions, i.e., involutions which do not fix any element.When we write F n , it is implied that n is even.Let (F n ) m be the set of m-tuples of fixed-point-free involutions.We say Fix α = (α 1 , . . ., α m ) ∈ (F n ) m with no triple matchings.We show how to canonically define an (m, n)-structure set S α with associated B-tree local involutions α 1 , . . ., α m .After fixing a marking M of T m × T n , by Proposition 2.5 this also defines (up to conjugacy) an involutive BMW group Γ α ∈ BMW M (m, n) with structure set S α .
For each 1 ≤ k < l ≤ n, set I k,l ∶= {i α i (k) = l}.Note that since α has no triple matchings, I k,l ≤ 2. The structure set S α is the collection of subsets {a i , b k , a j , b l } such that 1 ≤ i ≤ j ≤ m and 1 ≤ k < l ≤ n satisfy I k,l = {i, j} (note that i could equal j).It is straightforward to check that S α is indeed a structure set and that the B-tree local involutions of S α are exactly α 1 , . . ., α m .The structure set associated to α is then See Figure 3.
A random element of F n is an element of F n chosen uniformly at random.A random element of (F n ) m is an element of (F n ) m chosen uniformly at random, i.e., an m-tuple of m independently chosen, random elements of F n .We are now ready to define random involutive BMW groups: Definition 5.2.Suppose a marking M for T m ×T n is fixed.Let n > 0 be even and let α be a random element of (F n ) m .If α has no triple matchings, we define the corresponding random involutive BMW group of degree (m, n) to be Γ α ∈ BMW M (m, n).On the other hand, if α contains a triple matching, then we say that the corresponding random involutive BMW group is not defined.
Let P be a property of BMW groups.We say that a random involutive BMW group of degree (m, n) satisfies property P with probability p, if given a random α = (α 1 , . . ., α m ) ∈ (F n ) m , then with probability p the corresponding random involutive BMW group is defined and satisfies property P. n) has no triple matchings (and consequently defines a random BMW group) with probability tending to 1 as n tend to infinity.
Lemma 5.4.For n even, let O 1 , . . ., O k be a collection of unordered pairs of distinct elements in {1, . . ., n} such that O i ∩O j = ∅ for i ≠ j.The probability that a random element of F n contains the orbit The remainder of this section is devoted to showing that a random α ∈ F m n satisfies the three conditions from Proposition 6.2 with sufficiently high probability.Condition (A1) was shown to hold in Lemma 5.5.We thus begin with condition (A2).
Next, we give the probability that two fixed-point-free involutions share a common orbit.
Lemma 6.4.The probability that two random elements of F n share a common orbit is Moreover, this probability converges to 1 − e − 1 2 as n → ∞.
Proof.Suppose α, α ′ ∈ F n are chosen uniformly at random.Let {O 1 , . . ., O r } be the set of orbits of α where r = n 2 .Let T i ⊆ F n be the set of fixed-point-free involutions with orbit O i , and let Z be the number of fixed-point-free involutions in F n with orbit O i for some 1 ≤ i ≤ n.By inclusion-exclusion: By the above equation, we then have: By Lemma 5.3, we can divide by (n − 1)!! to conclude the first claim.We now prove the convergence claim.Set a k,r = (−1) k+1 r k and a k,r = 0 otherwise.We want to show that lim r→∞ ∑ r k=1 a k,r is equal to 1 − e − 1 2 .We first note that for all k ≤ r, a k,r = (−1) k+1 r k Corollary 6.5.There exists a number N such that whenever n ≥ N , the probability that two random elements in F n share a common orbit is at least 1 3 .
Remark 6.6.It can be shown using estimates that N in the previous corollary, can be taken to be 2.
Finally, we show that the third property of Proposition 6.2 holds with high probability.
Lemma 6.7.Let N be as in Corollary 6.5, n ≥ N and α ∈ (F n ) m be a random element.Then with probability at least 1 − 2m 2 8 9 m the following property holds: for all 1 ≤ i < i ′ ≤ m there exists a j such that α i and α j share a common orbit, and α j and α i ′ share a common orbit.
Proof.Let α = (α 1 , . . ., α m ).For each 1 ≤ i < i ′ ≤ m and j ≠ i, i ′ , let Y i,j,i ′ be the event that both α i and α j share a common orbit, and α j and α i ′ share a common orbit.Since (up to conjugation) we can treat α j as fixed, and α i and α i ′ as independently randomly chosen, we get that the event that α i and α j share an orbit and the event that α j and α i ′ share an orbit are independent.By Corollary 6.5 each of them occurs with probability at least 1 3 .Therefore, the probability of the event Y i,j,i ′ is P(Y i,j,i ′ ) ≥ 1 3 2 = 1 9 .Let Z i,i ′ be the event that no Y i,j,i ′ occurs for any j ∉ {i, i ′ }, and let Z = ⋃ i≠i ′ Z i,i ′ .Observe that the property in the lemma's statement holds exactly when the event Z does not occur.Since Z i,i ′ = ⋂ j∉{i,i ′ } Y c i,j,i ′ and the events Y i,j,i ′ are independent, we have that and by a union bound that Proof of Theorem 6.1.Suppose first that n ≥ N where N is as in Corollary 6.5.By Lemmas 5.5, 6.3 and 6.7, and by a union bound, a random α ∈ F m n satisfies the three properties of Proposition 6.2 with probability at least Since n > m 5 , we can pick some constant C such that p(m, n) ≥ 1 − C m as required.Moreover, by choosing C large enough, we can guarantee that this holds for all n (not just for n ≥ N ).

B-tree local action
In this section we show that the B-tree local action of a random BMW group contains the alternating group Alt(n) asymptotically almost surely (Corollary 7.2).This follows from a generation result for random fixed-point-free involutions (Theorem 7.1).This theorem should be compared to those of Dixon [Dix69] and Liebeck-Shalev [LS96] which address generation results for random permutations and random involutions respectively.In fact, our proof closely follows that of Liebeck-Shalev.
and let α ∈ (F n ) m be a random element.The probability that the B-tree local action of the random BMW group Proof.By Lemma 5.5, the group Γ α is well-defined with sufficiently high probability.As we saw in §5, the B-tree local action is generated by α 1 , . . ., α m .The bound above now follows from the theorem.
To prove Theorem 7.1, we follow the same strategy as that of Liebeck-Shalev [LS96].Given a group G, we let M G denote the set of maximal, proper subgroups of G. Let N n ⊂ M Sym(n) be the set of maximal proper subgroups of Sym(n) which do not contain Alt(n).If Alt(n) ≤ ⟨α 1 , . . ., α m ⟩, then the permutations α 1 , . . ., α m are contained in some subgroup M ∈ N n .It follows by a union bound that where α ∈ F n is chosen uniformly at random.
Given a group G, Liebeck-Shalev define the function as n → ∞ for all s > 1.By Lemma 7.3 below and (7.1) we have that: for some constant c.Theorem 7.1 then follows from the last two equations.Thus, we now turn our attention to proving the following: Lemma 7.3.For every > 0 there exists a constant c such that given any even integer n > 0, any M ∈ N n and a random α ∈ F n , then Proof.We follow the same outline as the proof of [LS96, Theorem 5.1].We have the following two equations 1)) each following from Stirling's approximation, where for the second equation we also use the identity Thus, there exists a constant c 0 ≥ 1 such that Additionally, since (2πn) we may also assume that c 0 satisfies We now consider three cases depending on the structure of the subgroup M .
Case 1: M is primitive.It is shown in [PS80] that every primitive subgroup M ∈ N n satisfies M ≤ 4 n .Furthermore, there exists a constant c 1 such that 4 n ≤ c 1 Sym(n) 2 .By (7.3): Case 2: M is not transitive.In this case, M can be identified with Sym(k)×Sym(l) for some k, l < n such that k + l = n.Furthermore, M ∩ F n = F k × F l if both k and l are even, and M ∩ F n = 1 otherwise.By (7.2) we get that where the third inequality follows since n ≤ 2πkl.
Case 3: M is transitive and imprimitive.
Fixing an involution τ ∈ Sym(l) with m transpositions, the number of fixedpoint-free involutions α ∈ M ∩ F n which can be written as α = (π 1 , . . ., π l ) ⋅ τ is where the first inequality follows from (7.3).We thus get the bound: M ∩ F n ≤ c l 0 (k!) l 2 ⋅ I l where I l is the set of all involutions in Sym(l).It is shown in [Mos55] that Therefore, by Stirling's approximation, there exists c 2 such that I l ≤ c 2 (l!) 1 2 for all l.From the last two equations we deduce that: By (7.2) and as M = (k!) l l!, we get Recall that M = Sym(k)≀Sym(l) is a maximal subgroup of Sym(n) that preserves a partition of n elements into l subsets of size k.Without loss of generality, let the n-element set be {1, . . ., k} × {1, . . ., l} and suppose that M preserves the partition ⊔ l j=1 {1, . . ., k} × {j}.

Irreducibility of random BMWs
The aim of this section is to complete the proof of Theorem B. More precisely, we prove the following: Theorem 8.1.There is a constant C such the following holds.If Γ is a random BMW involution group of degree (m, n) with n > m 5 , then all of the following hold with probability at least 1 − C m : (1) the A-tree local action of Γ is Sym(m); (2) the B-tree local action of Γ is either Sym(n) or Alt(n); (3) Γ is irreducible; (4) Γ is hereditarily just-infinite.
Conclusions (1) and (2) follow directly from Theorem 6.1 and Corollary 7.2 respectively.Moreover, conclusion (4) follows from conclusions (1)-(3) and Theorem 2.8.Thus, we are left to prove conclusion (3) regarding the irreducibility of Γ.By a theorem of Caprace [Cap, Theorem 1.2(vi)], conclusion (3) is implied by conclusions (1) and (2) as long as Thus, in order to finish the proof Theorem 8.1, it is enough to show that (3) holds whenever n is one of the values above.To do so, we actually show conclusion (3) holds whenever n > m 8 (covering the above finite cases) by using the theorem of Trofimov-Weiss stated below.Suppose that Γ is a group acting vertex-transitively on a locally finite connected graph X.We do not assume the action is faithful.Given a vertex v ∈ X, let Γ v be its stabilizer, and let Γ for some v ∈ V (X), then the image of the action Γ → Aut(X) is not discrete.
Recall that a BMW group Γ ≤ Aut(T m ) × Aut(T n ) acts on T m by projecting to the first factor.We prove Theorem 8.1 (3) by showing that the hypotheses of Theorem 8.2 are satisfied with sufficiently high probability for a random BMW group.We do this by investigating the following graph: Definition 8.3.Given α ∈ (F n ) m , define the following simplicial graph G α whose edges are colored black and white such that: • The vertex set of G α is B n = {b 1 , . . ., b n }.
• Vertices b i and b j are joined by an edge if there is some 1 By a slight abuse of notation, we identify each b ∈ B n with the element Γ that interchanges the endpoints of the edge incident to o and labeled by b.We determine the action of b on T m as follows.Let L be a path in T m × {o B } starting at o. Let a i1 , . . ., a ir ∈ A m be the labels of consecutive edges of L. Now let R be the unique 1 × r rectangle in T m × T n whose bottom left vertex is o, whose left edge is labelled by b and whose top edge is the path bL.Such a rectangle is shown in Figure 5, with a i ′ j and b kj as indicated in Figure 5.To prove Theorem 8.1, we need to show that conditions (Irr1)-(Irr5) are satisfied with high probability.This has already been established for (Irr1), (Irr3) and (Irr5) in Lemma 5.5, Corollary 7.2 and Theorem 7.1 respectively.Hence, all that remains is to show that conditions (Irr2) and (Irr4) hold with high probability.To do so, we investigate the following random variable.By Corollary 6.5 there is a constant N such that for n ≥ N , two random elements of F n do not have a common orbit with probability at most 2 3, i.e., P(M n,2 = 0) ≤ The result now follows from Remark 8.6.
We now show the following: Proposition 8.8.Let m, n ∈ N with n > m 8 and n even.Then with probability at least 1 − 1 m , for some b ∈ B n , the closed ball N 6 (b) ⊂ G α only contains white edges.We first prove some lemmas that are used in the proof of Proposition 8.8.Lemma 8.9.For any even n, E(M n,2 ) = n 2(n−1) ≤ 1.
Proof.Let (α, α ′ ) ∈ (F n ) 2 be a random element.As in Lemma 6.4, we may assume α is fixed and α ′ is chosen at random.Let C 1 , . . ., C r be the orbits of α, with r = n 2 , and let Y i be the indicator random variable associated to the event that C i is an orbit of α ′ .Then M n,2 = ∑ vertices that are at a distance of 5 or less from the endpoint of a black edge.Hence, as m 8 < n, the closed ball N 6 (b) contains no black edge for some vertex b.
Proof of Theorem 8.1.As noted in the paragraph after Theorem 8.1, we may assume n > m 8 .Lemma 5.5, Proposition 8.8, Corollary 7.2, Lemma 8.7 and Theorem 6.1 each give upper bounds for the probability than one of the conditions (Irr1)-(Irr5) in Lemma 8.4 do not hold.Taking a union bound, we see that there is a constant N such that if n ≥ max(m 8 , N ), the probability that at least one of (Irr1)-(Irr5) is not satisfied is at most for some constants C 1 and C 2 .Therefore for some sufficiently large constant C, we deduce that for all even n > m 8 conditions (Irr1)-(Irr5) of Lemma 8.4 are satisfied with probability at least 1 − C m .Lemma 8.4 ensures that the group Γ α satisfies conclusion (3) of Theorem 8.1 with probability at least 1 − C m .Theorem 8.1 now follows from the discussion immediately after its statement.
one can assign to it the closed (possibly degenerate) path of length 4 in K m,n connecting the vertices a i , a j to the vertices b k , b l .In this way, one can think of an (m, n)-structure set as a partition of the edges of the complete bi-partite graph on the vertices A m ⊔ B n into closed paths of length 4 such that each edge belongs to exactly one such path.
local action on T m (resp.on T n ) as the conjugacy class of the local action at some vertex of T m (resp.T n ).Let us focus on the local action on T n .Let M be a marking of T m × T n with base vertex o.Let π A and π B be the projections of T m × T n to the first and second factors respectively.Let o B = π B (o) ∈ T n .Edges incident to o B in T n are labeled by elements of B n as follows: the label of an edge e incident to o B is the label of the unique edge e ′ of T m × T n incident to o such that π B (e ′ ) = e.Let Γ be an involutive BMW group of degree (m, n) with structure set S. The local action of Γ at the vertex o B can be identified with a subgroup of Sym(A m ) ≃ Sym(m).Recall that Γ is generated by the elements a 1 , . . ., a m , b 1 , . . ., b n .Observe that Stab Γ (o B ) = ⟨a 1 , . . ., a m ⟩, and thus the local action on T n is generated by the action of each one of a 1 , . . ., a m ∈ A m .Denote by

Figure 1 .
Figure 1.Determining the action of α i (k).There exists a unique edge e ′ ∈ T m ×T n labeled b k and incident to o.The element a i acts on T m × T n by mapping o to the endpoint o ′ of the unique edge e 1 labeled a i incident to o.Since Γ preserves the Γ-induced labeling, a i e ′ is the unique edge e 2 of T m × T n incident to o ′ labeled b k .The edges e 1 , e 2 are adjacent edges of a (unique) square in T m × T n .Let e 1 , e 2 , e 3 , e 4 be the edges of this square as shown in Figure1.By the definition of S, their respective Γ-induced labels are a i , b k , a j , b l .We get that a i e = π B (a i e ′ ) = π B (e 2 ) = π B (e 4 ).Since e 4 is incident to o, the label of a i e is the same as that of e 4 , namely b l .We call the involutions α 1 , . . ., α m ∈ Sym(n) the B-tree local involutions.Similarly, we can define the A-tree local involutions β 1 , . . ., β n ∈ Sym(m) corresponding to the local actions of b 1 , . . ., b n on the tree T m .

Lemma 4. 3 .Figure 2 .
Figure 2. The partial structure set S 0 gives rise to a partial partition of the complete bipartite graph K m,n in the sense of Remark 2.3, each set in S 0 corresponds to a 4-cycle in K m,n contained in the associated highlighted subset in the figure.

v
be the pointwise stabilizer of the set of all vertices distance i or less from v. Recall that the local action of Γ is the subgroup of Sym(m) induced by the action of Γ v on the edges adjacent to v. The following is a consequence of a theorem of Trofimov-Weiss, as reformulated by Caprace [Cap19, §4.5].Theorem 8.2 ( [TW95, Theorem 1.4]).Suppose that a group Γ acts vertex transitively on a connected locally finite graph X with 2-transitive local action.If Γ

Figure 4 .
Figure 4.The graph G α for the permutations in Example 5.1.Bold edges represent 'black' edges and dotted edges represent 'white' edges.

r b = b k0 b k1 b k2 b kr−1 b kr bo o Figure 5 .
Figure 5. Determining the action of b ∈ B n on the A-tree T m .Let L ′ be the path in T m × {o B } corresponding to the bottom of the rectangle R, i.e.L ′ is the unique edge path starting at o and whose edges have labels a i ′ 1 , . . ., a i ′ r .

Definition 8. 5 .
Let m, n ∈ N with n even.Given α ∈ F n , defineP α ∶= {{i, α(i)} 1 ≤ i ≤ n}.For random α = (α 1 , . . ., α m ) ∈ (F n ) m , define the random variable M n,m (α) ∶= i<j P αi ∩ P αj .Remark 8.6.The number of black edges in G α is at most M n,m (α), with equality precisely when α has no triple matchings.In particular, G α has no black edges if and only if M n,m (α) = 0.The following proposition demonstrates that G α has at least one black edge with sufficiently high probability: Lemma 8.7.There exists a constant N such that given any m, n ∈ N with n even and n ≥ N and a random α ∈ (F n ) m , then the probability that G α has no black edge is at most2 3 m−1 .Proof.For random α = (α 1 , . . ., α m ) ∈ (F n ) m ,define the random variable Y n,m (α) = ∑ m i=2 P α1 ∩ P αi .It follows from the definition of M n,m that, for each α ∈ (F n ) m , 0 ≤ Y n,m (α) ≤ M n,m (α).Therefore P(M n,m = 0) ≤ P(Y n,m = 0).

2 3 .
Since Y n,m is the sum of m − 1 non-negative independent identically distributed random variables with the same distribution as M n,2 , we have for n ≥ N thatP(M n,m = 0) ≤ P(Y n,m = 0) ≤ 2 3 m−1 r i=1 Y i , and P(Y i = 1) = 1 n−1 by Lemma 5.4.Therefore, by the linearity of expectation, E(M n,2 ) = n 2(n−1) ≤ 1 as required.Lemma 8.10.For any A > 0,P(M n,m ≥ A) ≤ m 2 2A .Proof.Observe that M n,m is a sum of m(m−1) 2 random variables, each having the same probability distribution as M n,2 .By Lemma 8.9 and linearity of expectation, we see thatE(M n,m ) ≤ m(m−1) 2 ≤ m 22 .The result now follows by applying Markov's inequality.Proof of Proposition 8.8.By Lemma 8.10, P(M n,m ≥ m 3 2 ) ≤ 1 m .To prove Proposition 8.8, it thus suffices to show that if M n,m (α) < m 3 2 , then all edges in the closed ball N 6 (b) are white for some vertex b.Indeed, if M n,m (α) < m 3 2 then by Remark 8.6, G α contains fewer than m 3 2 black edges.Since vertices of G α have valence at most m, any edge of G α has at most 2m 5 vertices a distance 5 or less from it.Thus there are at most 2m 5 × m 3 2 = m 8 ).Finally, as S BR , S B , S M , S C1 , S C2 ⊂ S, α m−1 is the transposition (4, n − 2).From this we then conclude that α 1 , . . ., α m generate Sym(n).