Permutation-based presentations for Brin's higher-dimensional Thompson groups $nV$

The higher-dimensional Thompson groups $nV$, for $n \geq 2$, were introduced by Brin in 2005. We provide new presentations for each of these infinite simple groups. The first is an infinite presentation, analogous to the Coxeter presentation for the finite symmetric group, with generating set equal to the set of transpositions in $nV$ and reflecting the self-similar structure of $n$-dimensional Cantor space. We then exploit this infinite presentation to produce further finite presentations that are considerably smaller than those previously known.


Introduction
A well-known result of Brouwer [6] states that a non-empty totally disconnected compact metrizable space without isolated points is homeomorphic to the Cantor space C. Consequently this space arises throughout mathematics and it is unsurprising that many groups occur among its homeomorphisms.Interesting and important examples of such groups include Grigorchuk's group of intermediate growth [12,13], which may be naturally described as consisting of certain automorphisms of a binary rooted tree, and various generalizations, such as the Gupta-Sidki groups [15] and the multi-GGS groups (see, for example, [10]); the asynchronous rational group of Grigorchuk, Nekrashevych and Sushchanskii [14]; and, particularly relevant to this paper, the groups F , T and V introduced by Richard J. Thompson [7,21].
Thompson's group F is a 2-generator group with abelianization isomorphic to a free abelian group of rank 2 and such that its derived subgroup F ′ is simple.The other two groups T and V introduced by Thompson are both infinite simple groups.All three groups are finitely presented, with F having a small presentation with two generators and two relations.The presentations for T and V , as described in [7], both involve additional generators and relations to supplement those used for F .In particular, Thompson's original presentation for his group V involved four generators and fourteen relations.In work by Bleak and the author [2], we returned to possible presentations for V .We give there various presentations for this group: one involving infinitely many generators and an infinite family of relations.The generators in [2,Theorem 1.1] correspond to transpositions of certain disjoint basic open sets of Cantor space and the relations are analogous to the Coxeter presentation for a finite symmetric group, but also include what we termed "split relations" reflecting the self-similar structure of Cantor space.The second presentation, given in [2, Theorem 1.2], is a finite presentation, essentially obtained by reducing the infinite presentation, with three generators and eight relations (which compares favourably in size to Thompson's original presentation).We then produced a two-generator presentation for V by use of Tietze transformations and our smallest presentation, obtained via computational methods, is on two generators and seven relations [2,Theorem 1.3].One should also note a link between our infinite presentation for V and the geometric presentations given by Dehornoy [8].
In 2004, Brin [3] introduced, for each positive integer n 2, an analogue of Thompson's group V that acts upon an n-dimensional version of Cantor space.He denotes this group by nV and, via the homeomorphism C n ∼ = C, this family provide us with further groups of homeomorphisms of Cantor space.Bleak and Lanoue [1] noted that two of these groups mV and nV are isomorphic if and only if m = n.Brin observes in his first paper that the group 2V is an infinite simple group, while in [5] he shows that all the groups nV are simple.In the latter argument, he makes considerable reference to the baker's maps of C n and notes that these maps can be expressed as a product of transpositions.This observation will be particularly relevant to our proof of Theorem 1.1 in Section 2 below.Furthermore, all these groups are finitely presented: In [4, Theorem 5], Brin observes that 2V has a finite presentation with 8 generators and 70 relations.This method was extended by Hennig and Matucci [17] to establish a finite presentation for the groups nV involving 2n + 4 generators and 10n 2 + 10n + 10 relations (see [17,Theorem 25]).Indeed, each group nV is of type F ∞ , as established in [19,11].In this article, it is demonstrated that these groups possess infinite presentations involving elements corresponding to transpositions of disjoint basic open sets and involving relations that have a Coxeter-like shape and reflect the self-similar nature of C n (see Theorem 1.1 below).Again these infinite presentations bear comparison with Dehornoy's geometric presentations [8] for F and V .In the final section of the paper, we demonstrate that the group nV is isomorphic to a group G with a finite presentation involving 3 generators and 2n 2 + 3n + 11 relations.It is noteworthy that the number of generators is bounded independent of the parameter n and that the number of relations significantly improves upon the presentations in [4,17].In particular, the resulting finite presentation for 2V involves 3 generators and 25 relations.

Notation
We write C for the Cantor set; that is, the collection of all infinite words from the alphabet {0, 1}.We also use the set {0, 1} * of finite words in this alphabet and write ε to denote the empty word.If α, β ∈ {0, 1} * , then α β indicates that α is a prefix of β.On the other hand, the notation α ⊥ β denotes that α β and α β and we then say these words are incomparable.The length of a finite word α, denoted by |α|, is the number of symbols from {0, 1} occurring in α.
If n is a positive integer with n 2, the higher-dimensional Thompson group nV is defined (see below) as consisting of certain transformations defined on n-dimensional Cantor space Γ = C n .Accordingly, we shall also need the set Ω of sequences α = (α 1 , α 2 , . . ., α n ) where each α i ∈ {0, 1} * and we use the term address to refer to elements of Ω.These addresses are used to index the basic open subsets of Γ which in turn appear in the definition of the elements of nV .We extend the concept of incomparability to addresses by writing α ⊥ β, for a pair of addresses α = (α 1 , α 2 , . . ., α n ) and β = (β 1 , β 2 , . . ., β n ), when α d ⊥ β d for some index d with 1 d n.Similarly, for such addresses, we write α β when α d β d for all d = 1, 2, . . ., n.
The higher-dimensional Thompson group nV consists of certain homeomorphisms of Γ.We shall use right action notation throughout and so write wg for the image of w ∈ Γ under g ∈ nV .Let Γ(α) = { αw | w ∈ Γ } be the collection of all sequences in Γ with the address α as prefix and this is the basic open set indexed by α.Note Γ(α) ∩ Γ(β) = ∅ if and only if α ⊥ β and that Γ(α) ⊇ Γ(β) if and only if α β.An element g of nV is then described as follows: Given ) into the same number of disjoint basic open sets, we define the homeomorphism g of Γ by α (i) w → β (i) w for i = 1, 2, . . ., k and any w ∈ Γ.Thus each homeomorphism in nV is given by piecewise affine maps on Γ determined by two partitions of the space into the same number of basic open sets and some bijection between the parts.Figure 1 illustrates an example partition of C 3 ; that is, a potential choice for domain or codomain partially determining an element of 3V .If α and β are incomparable addresses, we call the element of nV that maps αw → βw, βw → αw for all w ∈ Γ and fixes all other points in Γ a transposition.This element has the effect of interchanging the basic open sets Γ(α) and Γ(β).In what follows, we shall write G ∞ for the group with the presentation given in Theorem 1.1 below.The element denoted by (α β) that appears in that presentation corresponds, under the natural homomorphism G ∞ → nV , to this transposition that interchanges Γ(α) and Γ(β).
To describe an address in Ω in theory requires one to write a sequence of n finite words in {0, 1}.Such a sequence would appear quite cumbersome in our calculations particularly when appearing as entries in the transpositions that we work with.Accordingly, we present a more compact and useful notation.If α is some (usually explicit) finite word in {0, 1}, we shall write α d for the address all of whose entries are the empty word with the exception of the dth coordinate which equals α.Thus, for example, 010 d = (ε, . . ., ε, 010, ε, . . ., ε) where 010 occurs in the dth coordinate in this n-tuple.We shall particularly make use of this notation when we wish to append one (or more) letters from {0, 1} to particular entries in an address α = (α 1 , α 2 , . . ., α n ).For example, we write α.0 d to indicate that we concatenate the addresses α and 0 d ; that is, we append the symbol 0 to the dth coordinate α d of α: (The use of the dot appearing this notation is to demarcate the end of the first address α and the beginning of the second and is intended to achieve clarity.Indeed, according to our notation, α0 d (without the dot) would indicate the address with a single non-empty entry α0 in the dth coordinate.The dot notation is unnecessary when concatenating two finite words in {0, 1} but helps when dealing with n-tuples.)The use of this notation can be observed within what we term the "split relations" appearing in the statement of Theorem 1.1 below (see Equation ( 4)) and in the addresses labelling the parts in Figure 1.An additional piece of notation that we shall use is that if x ∈ {0, 1}, then x denotes the other element in this set and then, following our above convention, xd is the sequence (ε, . . ., ε, x, ε, . . ., ε) where x occurs in the dth coordinate.Finally, ε will denote the address (ε, ε, . . ., ε) all of whose entries are the empty word.
To specify the relations that define our group, we define an additional notation that encodes the partial action of transpositions in nV upon the basic open sets indexed by the addresses in Ω.To be specific, if α, β, γ ∈ Ω with α ⊥ β, we define a partial map by Thus we associate to the symbol (α β) the partial map on the set Ω of addresses that performs a prefix substitution that interchanges the prefix α with the prefix β.

Statement of results
In Section 2, we shall establish the following infinite presentation for nV .
Theorem 1.1 Let n 2. Let A be the set of all symbols (α β) where α and β are addresses in Ω with α ⊥ β.Then Brin's higher-dimensional Thompson group nV has infinite presentation with generating set A and all relations where α, β, γ and δ range over all addresses in Ω such that α ⊥ β, γ ⊥ δ and such that both α • (γ δ) and β • (γ δ) are defined, and d ranges over all indices with 1 d n.
We shall refer to relations of the form (3) as "conjugacy relations" and those of the form (4) as "split relations" in what follows.The latter arise due to the self-similar nature of Cantor space: to exchange prefixes α and β is equivalent to exchanging both the pairs of prefixes obtained by "splitting" the dth coordinate.Note that we use exponential notation for conjugation writing g h for h −1 gh where g and h belong to some group and this is consistent with our use of right actions.
We shall also need an additional relation that can be deduced immediately from (3).On the face of it, if α and β are incomparable addresses in Ω, the symbols (α β) and (β α) are not necessarily the same element of the group with the given presentation.However, we expect them to correspond to the same transposition in nV .The "symmetry relation" follows from (3) by taking γ = α and δ = β: We shall make use of this additional relation (5) throughout our arguments.
The method of proof of the above theorem is essentially to verify a family of relations for nV originally found in [17].Let G ∞ denote the group with presentation given in Theorem 1.1.The key steps in the proof in Section 3 are to define and investigate elements in G ∞ that correspond to baker's maps.The two-dimensional baker's map is a basic object within the study of dynamical systems (see, for example, [9]) and is illustrated in Figure 2(i).In the context of n dimensions, we shall refer to baker's maps that arise in the domain from a cut in the first coordinate and in the codomain from a cut in the dth coordinate.Thus we define, in G ∞ , an element B d (α) that corresponds to the element of nV with support equal to Γ(α) mapping C n → C n via the formula When we refer below to the element B d (α) evaluating to an "index d" baker's map, we mean that it evaluates to the homeomorphism of C n given by this formula.All baker's maps arising within our work will have such a form (for some address α ∈ Ω and some d 2).The elements B d (α) in G ∞ are defined via the formulae expressing baker's maps in terms of transpositions found in [5].In Section 2, we observe that the behaviour of baker's maps can be deduced from the relations assumed about transpositions.Lemmas 2.2-2.4 give the properties upon which we depend.In summary, while Brin [5] establishes simplicity of nV by expressing baker's maps as products of transpositions, we use relational properties between transpositions to produce information about baker's maps to establish our presentation in Theorem 1.1.
In Section 3, we reduce our infinite presentation to a finite presentation (the relations are those listed in R1-R7 in that section): Theorem 1.2 Let n 2. Brin's higher-dimensional Thompson group nV has a finite presentation with three generators and 2n 2 + 3n + 11 relations.
To prove this theorem, we begin with transpositions with entries from the set ∆ of addresses α = (α 1 , α 2 , . . ., α n ) with |α d | = 2 for 1 d n.Thus, as a base point in an induction argument we assume that we have a subgroup isomorphic to (a quotient of) the symmetric group of degree 4 n .In an induction argument, we build further transpositions by successively conjugating the transpositions constructed at a previous stage and then finally exploit the split relations (4) to complete the definitions.In this way, we demonstrate that the group G with the presentations provided in Theorem 1.2 is a quotient of the group G ∞ described in Theorem 1.1.
Finally, by applying Tietze transformations, we shall deduce: Corollary 1.3 Let n 2. Brin's higher-dimensional Thompson group nV has a finite presentation with two generators and 2n 2 + 3n + 13 relations.
Remarks: In common with the finite presentation given by Hennig-Matucci [17], the number of relations we use is quadratic in the dimension n.One could ask whether there are presentations for nV on two generators, but where the number of relations grows at most linearly in n?In our case, the quadratic function arises from the family of Relations R5 which is used to ensure that the well-definedness of transpositions (α β) where two coordinates of α have length 3 and all remaining coordinates of α and all those of β have length 2. Although the growth in the number of relations relative to the dimension of the space acted upon seems reasonable, surprising results such as that of Guralnick-Kantor-Kassabov-Lubotzky [16] stand in contrast to expectations.The arguments used in [16] employ a process that has been termed the Burnside procedure and presented in detail in the Appendix of [20].This process can be used to reduce some large presentation of a group to a much smaller one.Potentially it could be applied to the infinite presentation found in Theorem 1.1 and one might wonder how the result would compare with Theorem 1.2.It seemed to the author that the most direct application of the Burnside procedure (if successful) would likely result in more relations.Nevertheless, it remains an interesting question whether smaller presentations exist for Brin's groups nV .

The infinite presentation for nV
We devote this section to establishing Theorem 1.1.Accordingly we define G ∞ to be the group presented by the generators A = { (α β) | α, β ∈ Ω, α ⊥ β } subject to the family of relations ( 2)- (4).In this context, we shall use the term transposition for any element (α β) appearing in the generating set A. It was observed by Brin [5] that the group nV is generated by the corresponding transpositions of basic open sets of Γ.It is readily verified that these homeomorphisms satisfy the relations listed in Theorem 1.1.Hence there exists a surjective homomorphism φ : G ∞ → nV that maps (α β) to the corresponding transposition in nV .In what follows, we shall speak of evaluating a product g in nV to mean the effect of applying the homomorphism φ to the element g ∈ G ∞ .We can extend the definition appearing in Equation ( 1) to a product g of transpositions, say g = g 1 g 2 . . .g k where each g i ∈ A, by defining α • g to equal the value obtained by successively applying Equation (1) with each transposition g i .Note that this is strictly speaking a function of the word in A representing g rather than depending upon g as an element of G ∞ .With this extended definition, if α, β ∈ Ω with α ⊥ β and g ∈ G ∞ is expressed as a product of transpositions in such that both α • g and β • g are defined, then it follows by repeated use of the conjugacy relations (3) that Note that α • g, when it is defined, coincides with the value obtained if the product g is evaluated as an element of the Brin's higher-dimensional Thompson group nV and then α • g is calculated via the natural partial action of nV upon the addresses Ω.The only difference is that there may exist some addresses α for which α•g is not defined for our given word representing g but for which the corresponding transformation in nV does have an action defined upon α.However, if α = (α 1 , α 2 , . . ., α n ) and provided the words α i are sufficiently long, then α • g is defined and hence coincides with the value obtained via the partial action of nV upon Ω.
We shall establish that G ∞ is isomorphic to the group nV by demonstrating that a family of relations found within Hennig-Matucci's work [17] can be deduced from our defining relations.We shall define elements of our group G ∞ that correspond to the family of generators that are used in [17].Some are readily constructed using transpositions (α β) but others depend upon building analogues of the baker's maps.Brin's paper [5] describes how to construct a baker's map from transpositions in his Lemma 3. By following this recipe we are able to define the required elements of G ∞ .Moreover, it then follows that the products of transformations we define evaluate to the required baker's maps in nV and hence we can determine the value of α • g for such products g provided the coordinates of α are sufficiently long.
If α, β ∈ Ω with α ⊥ β and d is an index with 2 d n, we define We define further elements of G ∞ in terms of this product as follows: where γ = (γ 1 , γ 2 , . . ., γ n ) is an address in Ω satisfying |γ 1 |, |γ d | 1 and the additional condition that γ ⊥ 0 1 in the first two definitions and that γ ⊥ 1 1 in the last two definitions in (7).Further, we then define: These three types of element are the analogues of the maps arising within the proof of [5, Lemma 3] and their definition precisely follows that proof.Consequently, the product A d (α, β) evaluates in the group nV to the composite of an "index d" baker's map with support Γ(α) and the inverse of an "index d" baker's map with support Γ(β).The subsequent elements Bd (α) and B d (α) both evaluate to the "index d" baker's map with support Γ(α).The difference is that the address α = (α 1 , α 2 , . . ., α n ) that we have first defined them upon satisfies |α 1 | = 1 for both products but the Bd version requires |α d | = 1 while B d permits α d to be empty.One notes that to define a baker's map on the whole space Γ = C n (that is, with address ε) requires a further such definition.As this is (up to choice of index d) a single element in G ∞ , we delay the definition of this element, which appears as C 0,d below (see Equation ( 11)).
To extend the baker's maps to arbitrary addresses we use another convenient notation.If g is an element of G ∞ and δ ∈ Ω, we write δ.g for the element of G ∞ obtained by inserting δ as a prefix in both entries of every transposition appearing in the product g.Since the relations ( 2)-( 4) are closed under performing such insertions, it follows that (i) δ.g is a well-defined element of G ∞ and (ii) if u = v is a relation that holds in G ∞ then δ.u = δ.v also holds in G ∞ .We shall use the latter observation to reduce the number of calculations required.
In terms of this prefix notation, observe that for any addresses α, β and δ with α ⊥ β.For our baker's maps, we define Bd (α), where α where Bd (β) is as defined in Equation (7).
(iii) We expand the right-hand side using the definition of the terms A d and collect the transpositions using the conjugacy relations (3): (by ( 4)) (ii) If α = δ.x 1 .yd for some δ ∈ Ω and some x, y ∈ {0, 1}, then If we take δ = ε in Part (ii) then it tells us that the definitions in Equation ( 7) are independent of the choice of address γ used.Consequently, Bd (α) does depend only on the address α.
Proof: (i) Repeatedly apply the split relation (4) to express (ζ η) as a product of transpositions (ζ ′ η ′ ) having entries with sufficiently long coordinates that the values ζ ′ • Bd (α) and η ′ • Bd (α) are defined.These values therefore coincide with those obtained when the corresponding baker's map in nV is applied to ζ ′ and η ′ .Since ζ and η are both incomparable with α, we conclude It then follows, by the conjugacy relation (3), that Bd (α) commutes with all such (ζ ′ η ′ ) and hence also with their product (ζ η).
(vi) We present the case x = y = 0, with the other cases established by similar calculations.Recall that γ ⊥ 0 1 in our definition (7) of Bd (0 1 .0d ).In the following calculation, we begin by applying Lemma 2.1(iii) to the terms appearing in the definition of Bd (0 1 .0d ): (vii) This follows by applying Lemma 2.1(iv) and the split relation (4) to terms appearing in the formula (7), and then rearranging in a similar way to the proof of Lemma 2.1(iv).
Proof: The first part of (i) is established by the same argument as used in Lemma 2.2(i) and the remainder follows immediately.Parts (ii) and (iii) follow using parts (iii) and (iv), respectively, of Lemma 2.2, while part (vi) is an extension of Lemma 2.2(vii) that is established similarly.
We establish part (iv) in the case that x = 0. First apply Lemma 2. We now consider one of the presentations for Brin's higher-dimensional Thompson group nV given by Hennig-Matucci [17].They define generators X m,d (for m 0 and 1 d n), C m,d (for m 0 and 2 d n), π m (for m 0) and πm (for m 0) and describe eighteen families of relations (numbered ( 1)-( 18) on pages 59-60 of [17]).They observe, in [17,Theorem 23], that these do indeed give a presentation for nV .Since we write our maps on the right, we shall convert the relations to our setting by reversing each one and record these now for reference.We have also changed some of the labels on Hennig-Matucci's generators appearing in the lists so that our arguments can be unified when establishing Proposition 2.5 below.In the families (HM1)-(HM7), one should assume that 1 d, d ′ n: for m < q, (HM1) X q,d π m = π m X q,d for q > m + 1, (HM4) X m,1 πm = πm+1 π m for m 0, (HM6) The second collection of relations is as below.Note that we have adjusted the range of the parameters in (HM8) to bring it into line with the relations given by Brin (see [4,Eqn. (22)]).
Finally, in the families (HM14)-(HM18), 2 d n and 1 d ′ n, unless otherwise indicated: We now define the elements of our group G ∞ that correspond to the above generators.In the following d is an index with 2 d n.First we set These are extended, for positive integers m 1, to By inserting the prefix 1 1 into the entries of transpositions in part (i) of this lemma, it follows with use of Lemma 2.3(iv) that: Proof: (i) First note that, by suitable choice of γ appearing in the definition (7), we can express C 0,d ′ as a product of transpositions whose entries have non-empty coordinates only for index 1 and index d ′ .Consequently, provided the index 1 and index d ′ coordinates of an address ζ are sufficiently long, ζ • C 0,d ′ is defined.Furthermore C 0,d ′ evaluates in nV to the (primary) baker's map with full support on Γ, so this value ζ • C 0,d ′ coincides with that obtained when we act with the baker's map.Now if x ∈ {0, 1}, then (x 1 .ζ)• C 0,d ′ is also defined (as the required coordinates are sufficiently long) and equals x d ′ .ζ in view of how the baker's map acts.If (α β) is any transposition, apply (4) repeatedly to express it as a product of transpositions (ζ η) with the index 1 and index d coordinates of ζ and η sufficiently long that the partial action of C 0,d ′ upon them is defined.By inserting x 1 as prefix into all the transpositions involved, we express (x 1 .αx 1 .β)as a product of transpositions (x by our above observation and similarly for x 1 .η,we deduce We now recombine the resulting transpositions using (4) to conclude Establishing this proposition completes the proof of Theorem 1.1 since it establishes that the surjective homomorphism φ : G ∞ → nV has trivial kernel.
Proof: First, as observed earlier, if u = v is a relation in G ∞ then so is 0 1 .u= 0 1 .v.Consequently, it suffices to establish each of (HM1)-(HM18) only when m = 0. Relations (HM8)-(HM13) are the most straightforward to verify and follow directly from the assumed relations involving transpositions (i.e., (2) and ( 3)).Relation (HM6) is established in a similar manner.When q > 1, observe that both X q,d and C q,d is a product of transpositions all of whose entries have 00 1 as a prefix.These transpositions are therefore disjoint from π 0 and relations (HM4) and (HM17) follow.We now describe the details involved in the other relations.
Note that 00 1 • X 0,1 = 0 1 .Hence, for any g ∈ G ∞ , This establishes (HM1), (HM2) and (HM5) in the case when m = 0 and d = 1, and it also establishes (HM15) in the case when m = 0 and d ′ = 1.We extend the first three to d 2 by use of Lemma 2.3(i) to tell us that B d (1 1 ) commutes with each of X q,d ′ , π q and πq for q > 0. Similarly, we extend (HM15) to the case when d ′ 2 by using the same fact to show B d ′ (1 1 ) commutes with C q,d for q 2 and, by Lemma 2.3(v), also with C 1,d .
The case when d 2 now follows using Lemma 2.3(i)-(iii).We establish Relation (HM7) first in the case when d ′ = 1 and m = 0.If d 2, then: We can interchange the roles of d and d ′ in (HM7) by multiplying on the left by π m+1 .Hence it remains to establish the relation in the cases when both d, d ′ 2. This is achieved as follows: Relation (HM14) is established by using formulae about conjugation by (0 1 1 1 ): For (HM16), we collect the transpositions comprising X 0,1 to the right: Finally consider relation (HM18) in the case when m = 0. We calculate as required.This completes the proof of the proposition and the work of this section.
We shall define a presentation for a group G on three generators a, b and c and 2n 2 + 3n + 11 relations.The starting point is a presentation for the symmetric group of degree 4 n on two generators.According to [16, Theorem A], this can be achieved using merely eight relations (independent of the value of n).A recent article by Huxford [18] presents corrections to the arguments and the relations given in [16].Note, however, that the number of relations is unaffected and the statement of [16, Theorem A] remains valid.
Our first family (R1 below) of relations is sufficient to ensure that the generators a and b satisfy all relations that hold in the symmetric group of degree 4 n .Moreover, all the relations that we assume are satisfied if one maps a, b and c to the corresponding elements of nV (where for c we interpret Relation R7 in nV ).In particular, the resulting homomorphism G → nV induces a homomorphism from H = a, b onto the above symmetric group.Consequently, H ∼ = Sym(∆) and we may interpret the elements in H as defining permutations of ∆.We therefore use the symbol (α β), where α, β ∈ ∆ with α ⊥ β, to denote certain elements of the subgroup H and more generally refer to permutations of ∆ by which we mean the corresponding elements of this subgroup.This also means that we can speak of the support of an element g ∈ H and use the notation γ • g to denote the effect of applying g to some address γ ∈ ∆.(In some sense, this extends the notation given in Section 1.) The third generator c will be used to construct further transpositions (α β) for other addresses α, β ∈ Ω * with α ⊥ β.The details of this construction will be described later in this section, together with appropriate verifications that the resulting elements of G are well-defined and satisfy the relations ( 2)-( 4) listed in Theorem 1.1.For the remaining discussion, prior to explaining the construction, we shall assume the existence of the various transpositions (α β), each of which will be expressed as some product in G involving the generators a, b and c.
We order weights lexicographically, so wt(α) < wt(β) means either m(α) < m(β), or m(α) = m(β) and k(α) < k(β); that is, either the coordinates of α are all shorter than the longest of β or that the greatest length is the same but α has fewer of these longest coordinates than β.
In each step of the induction, we assume that we already have defined transpositions whose entries have weight less than (m, k) and verified all Relations (2)-( 4), and also (5), involving such transpositions.Our first stage is then to define transpositions (α β) where wt(α) = (m, k) and β ∈ ∆, or vice versa.We then verify that our definitions make sense and that all the required relations involving the newly defined transpositions are satisfied.At the second stage, we perform the same definitions and verifications for the remaining transpositions (α β) with wt(α), wt(β) (m, k).The new transpositions will always be conjugates of transpositions from Sym(∆) and consequently Relation (2) will always hold and we do not verify it explicitly.
Proof: In order that α • (γ δ) be defined, we require (i) γ ⊥ α or γ α, and (ii) δ ⊥ α or δ α.A similar pair of conditions apply when β • (γ δ) is defined.We analyze the four resulting conditions.Furthermore, we note, for example, that if γ α or δ α, then by exploiting the symmetry (γ δ) = (δ γ) we may assume that in fact γ α.This reduces the four conditions to the configurations described in the statement of the lemma.
Part (i) of Lemma 3.3 establishes any instance of the conjugacy relation ( 3) when (γ δ) ∈ H.We consider now the remaining instances of (3) involving transpositions defined at this stage of the induction (via ( 16) above) and we split into the cases listed in Lemma 3.1.
(B): In view of Lemma 3.3(i) and the symmetry between γ and δ in the conjugacy relation, we may assume in this case that wt(γ) = (3, 1) and α = γ.The remaining addresses β and δ must be from ∆. Write γ = γ.xd for some x ∈ {0, 1} and some index d.Conjugate the equation by use of Lemma 3.2(i) and 3.3(ii).Finally conjugate by a permutation σ of ∆ moving δ (d) to γ, δ (n+1) to β and δ (n+2) to δ, using Lemma 3.3(i), to establish the required relation.
In order to achieve Relation (5), we also set (β α) := (α β).We must verify that the above definition is independent of the choice of the address ζ and of the index d.
With the above assumptions, we make the following observations: Proof: (i) As distinct addresses in ∆, certainly ζ and η are incomparable.All addresses appearing in the following calculation have weight < (m, k) and so, by induction, and the required equation then follows.
(ii) Note that our assumption that α ⊥ β and β ∈ ∆ implies that β ⊥ γ.Now consider first the case when (m, k) = (3, 2), so that γ ∈ ∆.We simply conjugate Relation R5 by a permutation σ ∈ Sym(∆) that moves δ (0) to ζ, δ (1) to β and δ (2) to γ (using relations established in the weight (3, 1) stage) to yield the required formula.Now consider the case when (m, k) > (3, 2).Choose δ ∈ ∆ that is incomparable with each of β, γ and ζ.Note that wt(δ.xd .yd ′ ) = (3, 2) and so the transpositions with this address as an entry in the following calculation were constructed at an earlier stage.Then It therefore follows that the left-hand sides of these equations are equal, from which we deduce our required formula.It follows from part (i) of this lemma that our definition ( 17) of (α β) is independent of the choice of ζ.Then part (ii) shows the definition is also independent of the choice of index d.In conclusion, the transpositions (α β), where wt(α) = (m, k) and β ∈ ∆ or vice versa, are well-defined.The remaining work in this part of the induction is to establish the four types of conjugacy relations (3) and then the split relation (4) when they involve such transpositions.
(A): Consider two transpositions (α β) and (γ δ), at least one of which was defined as in Equation ( 17) and the other possibly arriving at an early stage in the induction, such that every pair of addresses from {α, β, γ, δ} is incomparable.Exploiting the symmetry relation ( 5), we can suppose without loss of generality that one of the following sets of conditions holds: In Case (B.ii), note that there is an index d such that the dth coordinate of γη has length m and that of γ is shorter.Therefore we can write η = η.xd and choose ζ ∈ ∆ incomparable with each of β, γ and δ to define (γη For Case (B.iii), there are two possibilities.If there is some d such that the dth coordinate of γη and δη both have length m but those of γ and δ are shorter, then we use the same argument as for Case (B.ii), but now the last step in the calculation is actually the definition of (δη β).
Otherwise, there are d and d ′ such that the dth coordinate of γη has length m and that of γ is shorter and the d ′ th coordinate of δη has length m and that of δ is shorter.Moreover, by hypothesis, the dth coordinate of γ must be longer than that of δ, so has length at least 3. and similarly for (δη β) (as in the second set of calculations below).Furthermore wt(θη) < (m, k) since the dth coordinate of θη is shorter than that of γη.We therefore compute: Hence (γη β) (γ δ) (γ θ) = (δη β) and, with use of our already verified Type (A) conjugacy relation, we conclude (γη β) (γ δ) = (δη β) (γ θ) = (δη β).
(C): In the notation of Lemma 3.1(C), if it were the case that wt(γη) = wt(γθ) = (m, k), then at this stage δ, δη, δθ ∈ ∆, which would force η = θ = ε.The conjugacy relation would reduce to one form g g = g that holds in any group.Consequently, upon exploiting the symmetry in the relation, we must verify (3) in the following two cases: for incomparable addresses β, γ, δ ∈ Ω * and some (possibly empty) η ∈ Ω such that the addresses appearing in the formula all have weight (m, k).Choose distinct ζ, θ ∈ ∆ incomparable with each of β, γ and δ, so that (γη β) = (γη ζ) (β ζ) and (γ δ) = (γ θ) (δ θ) .In the following calculation, all the transpositions manipulated have second entry either ζ or θ (selected from ∆): for incomparable α, β ∈ ∆.This is deduced from Relation R6 by conjugating by a permutation σ ∈ Sym(∆) that moves δ (0) to α and δ (1) to β. ( Both transpositions on the right-hand side exist by our assumption.Furthermore, the transpositions on the right-hand satisfy the relations (2) and ( 5 where some entry here has its dth coordinate of length k d − 1.We may assume the entry with this shorter coordinate is either γ (and possibly also γη) or β.If the dth coordinate of γ has length k d − 1 and that of η is empty, then we use the formula (19) for both (γη β) and (γ δ).Note then η.x d = x d .ηfor x ∈ {0, 1}, which permits us to calculate the following conjugate: relying upon relations that hold by the inductive assumption.The last step is either one of these assumed relations or is the definition of (δη β) if it is the case that the dth coordinate of β or δ has length k d − 1. Alternatively if the dth coordinate of γ has length k d − 1 and that of η is non-empty, write η = x d .η for some x ∈ {0, 1} and some (possibly empty) η ∈ Ω.In this case, we use Equation (19) for the definition of (γ δ) and calculate  4) involving the transpositions defined in (19) are either simply that definition or are inherited from split relations for the terms on the right-hand side of that formula.It now follows, using this step repeatedly, that we have constructed transpositions (α β), for α, β ∈ Ω with α ⊥ β, in the group G and verified all relations (2)-( 4) involving these transpositions.Consequently, by Theorem 1.1, there is a homomorphism φ : nV → G mapping each transposition in nV to the corresponding element that we have defined in G.Moreover, Relation R7 tells us that the generator c is in the image of φ and hence G is isomorphic to a quotient of nV .On the other hand, all relations R1-R7 listed are satisfied by the corresponding elements of nV and so there is a homomorphism from G into nV with non-trivial image.The fact that nV is simple therefore yields G ∼ = nV , completing the proof of Theorem 1.2.

Proof of Corollary 1.3:
The subgroup H = a, b ∼ = Sym(∆) of G can be generated by a cycle x of length 4 n and a transposition t that can be assumed disjoint from c (as described via Relation R7).Note that c has odd order.Therefore c and t are powers of y = ct and {x, y} is a generating set for G. Applying Tietze transformations to produce a presentation on generators x and y introduces two additional relations.This establishes the corollary.
conjugate by (γ γ ′ ) and use part (i) to produce the required formula for Bd (δ.x 1 .yd ).If γ and γ ′ are not incomparable, then as |γ d |, |γ ′ d | 1 there is another address ζ, with non-empty dth coordinate, that is incomparable with all three of γ, γ ′ and δ.x 1 .Now conjugate by the product (γ ζ) (γ ′ ζ), again using part (i), to produce the required formula(10) involving the address γ ′ .Part (iii) follows immediately from the definition of the Bd elements.

Lemma 2 . 3
Let d, d ′ be indices in the range 2 d, d ′ n.
d for m 2 and 2 d n, (12) where 0 m 1 denotes 00 . . .0 1 = (00 . . .0, ε, . . ., ε) with the word 00 . . .0 having length m.Note that, by Lemma 2.3(iv), C 1,d = 0 1 .C 0,d .Consequently, the definition of C m,d can be extended to include the case m = 1; that is, C m,d = 0 m 1 .C 0,d for all m 1 and all indices d in the range 2 d n.This will enable us to treat these baker's maps in a uniform manner.Lemma 2.4 Let d and d ′ be indices in the range 2 d, d ′ n.

Proposition 2 . 5
then it is a product of transpositions and it follows from the above calculation that (x 1 .g)C 0,d ′ = x d ′ .g for any x ∈ {0, 1}.The claimed equations now follow by taking g = C 0,d and noting, by use Lemma 2.3(iv), that B d (x 1 ) = x 1 .C 0,d .Part (ii) is an extension of Lemma 2.3(vi) established by a similar argument.The elements X m,d , C m,d , π m and πm of G ∞ defined in Equations (11) and (12) satisfy the relations (HM1)-(HM18).