COMMENSURATIONS AND METRIC PROPERTIES OF HOUGHTON’S GROUPS

JOSÉ BURILLO, SEAN CLEARY, ARMANDO MARTINO, AND CLAAS E. RÖVER

Abstract. We describe the automorphism groups and the abstract commensurators of Houghton’s groups. Then we give sharp estimates for the word metric of these groups and deduce that the commensurators embed into the corresponding quasi-isometry groups. As a further consequence, we obtain that the Houghton group on two rays is at least quadratically distorted in those with three or more rays.

Introduction

The family of Houghton groups H_{n} was introduced by Houghton [7]. These groups serve as an interesting family of groups, studied by Brown [2], who described their homological finiteness properties, by Röver [10], who showed that these groups are all subgroups of Thompson’s group V, and by Lehnert [9] who described the metric for H_{2}. Lee [8] described isoperimetric bounds, and de Cornulier, Guyot, and Pitsch [4] showed that they are isolated points in the space of groups.

Here, we classify automorphisms and determine the abstract commensurator of H_{n}. We also give sharp estimates for the word metric which are sufficient to show that the map from the abstract commensurator to the group of quasi-isometries of H_{n} is an injection.

1. Definitions and background

Let ℕ be the set of natural numbers (positive integers) and n ≥ 1 be an integer. We write ℤ_{n} for the integers modulo n with addition and put R_{n} = ℤ_{n} × ℕ. We interpret R_{n} as the graph of n pairwise disjoint rays; each vertex (i, k) is connected to (i, k + 1). We denote by Sym_{n}, FSym_{n} and FAlt_{n}, or simply Sym, FSym and FAlt if n is understood, the full symmetric group, the finitary symmetric group and the finitary alternating group on the set R_{n}, respectively.

Date: January 20, 2015.

The authors are grateful for the hospitality of NUI Galway. The first author acknowledges support from MEC grant #MTM2011-25955. The second author acknowledges support from the National Science Foundation and that this work was partially supported by a grant from the Simons Foundation (#234548 to Sean Cleary).

The Houghton group H_{n} is the subgroup of Sym consisting of those permutations that are eventually translations (of each of the rays). In other words, the permutation σ of the set R_{n} is in H_{n} if there exist integers N ≥ 0 and t_{i} = t_{i}(σ) for i ∈ ℤ_{n} such that for all k ≥ N, (i, k)σ = (i, k + t_{i}); throughout we will use right actions.

Note that necessarily the sum of the translations t_{i} must be zero because the permutation needs of course to be a bijection. This implies that H_{1} ≅ FSym.

For i, j ∈ ℤ_{n} with i ≠ j let g_{ij} ∈ H_{n} be the element which translates the line obtained by joining rays i and j, given by

(i, n)g_{ij} = (i, n − 1) if n > 1,
(i, 1)g_{ij} = (j, 1),
(j, n)g_{ij} = (j, n + 1) if n ≥ 1 and
(k, n)g_{ij} = (k, n) if k ∉ {i, j}.

We also write g_{i} instead of g_{i i+1}. It is easy to see that {g_{i} | i ∈ ℤ_{n}}, as well as {g_{ij} | i, j ∈ ℤ_{n}, i ≠ j}, are generating sets for H_{n} if n ≥ 3 as we can simply check that the commutator [g_{0}, g_{1}] = g_{0}^{−1}g_{1}^{−1}g_{0}g_{1} transposes (1, 1) and (2, 1). In the special case of H_{2}, an additional generator to g_{0} is needed and we choose τ which fixes all points except for transposing (0, 1) and (1, 1).

It is now clear that the commutator subgroup of H_{n} is given by

H′_{n} = \{ \begin{matrix}FAlt, & if n ≤ 2 \\ FSym, & if n ≥ 3 \end{matrix}

For n ≥ 3, we thus have a short exact sequence

1 ⟶ FSym ⟶ H_{n} ⟶^{π} ℤ^{n−1} ⟶ 1

where π(σ) = (t_{0}(σ), …, t_{n−2}(σ)) is the abelianization homomorphism. We note that as the sum of all the eventual translations must be zero, we have the last translation is determined by the preceding ones:

(1)  t_{n−1}(σ) = − ∑_{i=0}^{n−2} t_{i}(σ).

We will use the following facts freely throughout this paper, see Dixon and Mortimer [5] or Scott [11].

Lemma 1.1. The group FAlt is simple and equal to the commutator subgroup of FSym, and Aut(FAlt) = Aut(FSym) = Sym.

2. Automorphisms of H_{n}

Here we determine the automorphism group of H_{n}. First we establish that we have to look no further than Sym. We let N_{G}(H) denote the normalizer, in G, of the subgroup H of G.

Proposition 2.1. Every automorphism of H_{n}, n ≥ 1, is given by conjugation by an element of Sym, that is to say Aut(H_{n}) = N_{Sym}(H_{n}).

Proof. From the above, the finitary alternating group FAlt is the second derived subgroup of H_{n}, and hence characteristic in H_{n}. So every automorphism of H_{n} restricts to an automorphism of FAlt. Since Aut(FAlt) = Sym, this restriction yields a homomorphism Aut(H_{n}) ⟶ Sym and we only need to show that it is injective.

In order to see this let α ∈ Aut(H_{n}) be an automorphism whose restriction to FAlt is trivial. We let k ∈ ℕ and consider the following six consecutive points a_{ℓ} = (i, k + ℓ) of R_{n} for ℓ ∈ {0, 1, …, 5}.

We denote by g_{i}^{α} the image of g_{i} under α, and by (x y z) the 3-cycle of the points x, y and z. Using the identities

g_{i}^{−1}(a_{1} a_{2} a_{3})g_{i} = (a_{0} a_{1} a_{2}) and g_{i}^{−1}(a_{3} a_{4} a_{5})g_{i} = (a_{2} a_{3} a_{4})

and applying α, which is trivial on FAlt, we get

(g_{i}^{α})^{−1}(a_{1} a_{2} a_{3})g_{i}^{α} = (a_{0} a_{1} a_{2}) and (g_{i}^{α})^{−1}(a_{3} a_{4} a_{5})g_{i}^{α} = (a_{2} a_{3} a_{4}),

which imply that g_{i}^{α} maps a_{3} to a_{2}. Applying a similar argument to all points in the branches i and i + 1, it follows that g_{i}^{α} = g_{i}, and since i was arbitrary, this means that α is trivial. □

With Lemma 1.1 in mind we now present the complete description of Aut(H_{n}).

Theorem 2.2. For n ≥ 2, the automorphism group Aut(H_{n}) of the Houghton group H_{n} is isomorphic to the semidirect product H_{n} ⋊ S_{n}, where S_{n} is the symmetric group that permutes the n rays isometrically.

Proof. It is clear that every isometric permutation of the rays, (that is, automorphism of the graph R_{n}), induces an automorphism of H_{n}. By the proposition, it therefore suffices to prove that every α ∈ Sym which normalizes H_{n} must map (i, k + m) to (j, l + m) for some k, l ≥ 1 and all m ≥ 0.

To this end, we pick α ∈ N_{Sym}(H_{n}) and σ ∈ H_{n}. Since σ^{α}(= α^{−1}σα) is in H_{n} and maps the point xα to (xσ)α, these two points must lie on the same ray for all but finitely many x ∈ R_{n}. Similarly, x and xσ lie on the same ray for all but finitely many x ∈ R_{n}, as σ ∈ H_{n}. Combining these two facts, we see that α maps all but finitely many points of ray i to one and the same ray, say ray j.

Notice that we have the freedom to choose σ such that x and xσ are neighbors for all but finitely many x on ray i. If x and xα were not neighbors on ray j for infinitely many x, then t_{j}(σ^{α}) > 1. However, there is simply is not enough room left on ray i so that α^{−1} can map all but finitely many points from ray j to ray i. So t_{j}(σ^{α}) = 1 and the result follows. □

3. Commensurations of H_{n}

First, we recall that a commensuration of a group G is an isomorphism A ⟶^{ϕ} B, where A and B are subgroups of finite index in G. Two commensurations ϕ and ψ of G are equivalent if there exists a subgroup A of finite index in G, such that the restrictions of ϕ and ψ to A are equal. The set of all commensurations of G modulo this equivalence relation forms a group, known as the (abstract) commensurator of G, and is denoted Com(G). In this section we will determine the commensurator of H_{n}.

For a moment, we let H be a subgroup of a group G. The relative commensurator of H in G is denoted Com_{G}(H) and consists of those g ∈ G such that H ∩ H^{g} has finite index in both H and H^{g}. Similar to the homomorphism from N_{G}(H) to Aut(H), there is a homomorphism from Com_{G}(H) to Com(H); Its kernel consists of those elements of G that centralize a finite index subgroup of H.

In order to pin down Com(H_{n}), we first establish that every commensuration of H_{n} can be seen as conjugation by an element of Sym, and then characterize Com_{Sym}(H_{n}).

Since a commensuration ϕ and its restriction to a subgroup of finite index in its domain are equivalent, we can restrict our attention to the following family of subgroups of finite index in H_{n}, in order to exhibit Com(H_{n}). For every integer p ≥ 1, we define the subgroup U_{p} of H_{n} by

U_{p} = ⟨FAlt, g_{i}^{p} | i ∈ ℤ_{n}⟩.

We collect some useful properties of these subgroup first, where A ⊂_{f} B means that A is a subgroup of finite index in B.

Lemma 3.1. (1) For every p, the group U_{p} coincides with H_{n}^{p}, the subgroup generated by all p^{th} powers in H_{n}, and hence is characteristic in H_{n}.

(2) U′_{p} = \{ \begin{matrix}FAlt, & p even \\ FSym, & p odd \end{matrix} and |H_{n}: U_{p}| = \{ \begin{matrix}2p^{n−1}, & p even \\ p^{n−1}, & p odd \end{matrix} .

(3) For every finite index subgroup U of H_{n}, there exists a p ≥ 1 with FAlt = U′_{p} ⊂ U_{p} ⊂_{f} U ⊂_{f} H_{n}.

Proof. First we establish (2). We know that [g_{i}, g_{j}] is either trivial, when j ∉ {i − 1, i + 1}, or an odd permutation. So the commutator identities [ab, c] = [a, c]^{b}[b, c] and [a, bc] = [a, c][a, b]^{c} imply the first part, and the second part follows immediately using the facts from Section 1.

Part (1) is now an exercise, using that FAlt^{p} = FAlt.

In order to show (3), let U be a subgroup of finite index in H_{n}. The facts that FAlt is simple and U contains a normal finite index subgroup of H_{n}, imply that FAlt ⊂ U. Let p be the smallest even integer such that (pℤ)^{n−1} is contained in the image of U in the abelianisation of H_{n}. It is now clear that U_{p} is contained in U. □

Noting that Com(H_{1}) = Aut(H_{1}) = Sym, we now characterize the commensurators of Houghton’s groups.

Theorem 3.2. Let n ≥ 2. Every commensuration of H_{n} normalizes U_{p} for some even integer p. The group N_{p} = N_{Sym}(U_{p}) is isomorphic to H_{n} ⋊ (S_{p} ≀ S_{n}), and Com(H_{n}) is the direct limit of N_{p} with even p under the natural embeddings N_{p} ⟶ N_{pq} for q ∈ ℕ.

Proof. Let ϕ ∈ Com(H_{n}). By Lemma 3.1, we can assume that U_{p} is contained in the domain of both ϕ and ϕ^{−1} for some even p. Let V be the image of U_{p} under ϕ. Then V has finite index in H_{n} and so contains FAlt, by Lemma 3.1. However, the set of elements of finite order in V equals [V, V], whence [V, V] = FAlt, as FAlt and FSym are not isomorphic. This means that the restriction of ϕ to FAlt is an automorphism of FAlt, and hence yields a homomorphism

ι: Com(H_{n}) ⟶ Com_{Sym}(H_{n}).

That ι is injective follows from a similar argument to the one in Proposition 2.1 applied to g_{i} ^{p} and six points of the form a_{ℓ} = (i, k + pℓ) with ℓ ∈ {0, 1, …, 5}. Since the centralizer in Sym of FAlt, and hence of any finite index subgroup of H_{n}, is trivial, the natural homomorphism from Com_{Sym}(H_{n}) to Com(H_{n}) mentioned above is also injective, and we conclude that Com(H_{n}) is isomorphic to Com_{Sym}(H_{n}).

From now on, we assume that ϕ ∈ Com_{Sym}(H_{n}). In particular, the action of ϕ is given by conjugation, and our hypothesis is that U_{p}^{ϕ} ⊂ H_{n}. Now we can apply the argument from the second paragraph in the proof of Theorem 2.2 to σ ∈ U and σ^{ϕ} (instead of σ^{α}), which only uses that both are elements of H_{n}. So ϕ maps all but finitely many points of ray i to one and the same ray, say ray j. The same argument shows that ϕ^{−1} maps all but finitely many points of ray j to ray i and we conclude that ϕ is a bijection between ray i and ray j after excluding finitely many points.

Since g_{i}^{p} has p infinite orbits intersecting ray i, the element (g_{i}^{p})^{ϕ} has, up to finitely many exceptions, p infinite orbits intersecting ray j. Since (g_{i}^{p})^{ϕ} is an element of H_{n}, we see that t_{j}((g_{i}^{p})^{ϕ}) = ±p. The same argument applies to ray i+1 and almost all of its image under ϕ which shows that (g_{i}^{p})^{ϕ} ∈ U_{p}. Since i was arbitrary and similar arguments hold for ϕ^{−1}, this means that ϕ normalises U_{p}.

This proves the theorem, where S_{p} ≀ S_{n} is the permutational wreath product of the symmetric group of degree p, which permutes the residue classes modulo p of one ray isometrically, and the symmetric group of degree n, acting as top group, permuting the n rays isometrically. □

We note that Com(H_{n}) is not finitely generated, for if it were, it would lie in some maximal N_{p}.

4. Metric estimates for H_{n}

In this section we will give sharp estimates for the word length of elements of Houghton’s groups. This makes no sense for H_{1} which is not finitely generated. As mentioned in the introduction, the metric in H_{2} was described by Lehnert [9]. In order to deal with H_{n} for n ≥ 3, we introduce the following measure of complexity of an element.

Given σ ∈ H_{n}, we define p_{i}(σ), for i ∈ ℤ_{n}, to be the largest integer such that (i, p_{i}(σ))σ ≠ (i, p_{i}(σ) + t_{i}(σ)). Note that if t_{i}(σ) < 0, then necessarily p_{i}(σ) ≥ |t_{i}(σ)|, as the first element in each ray is numbered 1.

The complexity of σ ∈ H_{n} is the natural number P (σ)), defined by

P (σ) = ∑_{i∈ℤ_{n}} p_{i}(σ).

And the translation amount of σ is

T (σ) = \frac{1}{2} ∑_{i∈ℤ_{n}} |t_{i}(σ)|.

The above remark combined with (1) immediately implies P (σ) ≥ T (σ). It is easy to see that an element with complexity zero is trivial, and only the generators g_{ij} have complexity one.

Theorem 4.1. Let n ≥ 3 and σ ∈ H_{n}, with complexity P = P (σ) ≥ 2. Then the word length |σ| of σ with respect to any finite generating set satisfies

P/C ≤ |σ| ≤ KP log P,

where the constants C and K only depend on the choice of generating set.

Proof. Since the word length with respect to two different finite generating sets differs only by a multiplicative constant, we can and will choose { g_{ij} | i, j ∈ ℤ_{n}, i ≠ j} as generating set to work with, and show that the statement holds with C = 1 and K = 7.

The lower bound is established by examining how multiplication by a generator can change the complexity. Suppose σ has complexity P and consider σg_{ij}. It is not difficult to see that

(2) p_{k}(σg_{ij}) = \{ \begin{matrix}p_{k}(σ) + 1, & if k = i and (i, p_{i}(σ) + 1)σ = (i, 1) \\ p_{k}(σ) − 1, & if k = j, (j, p_{j}(σ) + 1)σ = (j, 1) and \\ & (j, p_{j}(σ))σ = (i, 1) \\ p_{k}(σ), & otherwise \end{matrix}

where the first two cases are mutually exclusive, as i ≠ j. Thus |P (σg_{ij}) − P (σ)| ≤ 1, which establishes the lower bound.

The upper bound is obtained as as follows. Suppose σ ∈ H_{n} has complexity P. First we show by induction on T = T (σ) that there is a word ρ of length at most T ≤ P such that the complexity of σρ is P̄ with P̄ ≤ P and T (σρ) = 0. The case T = 0 is trivial. If T > 0, then there are i, j ∈ ℤ_{n} with t_{i}(σ) > 0 and t_{j}(σ) < 0. So T (σg_{ij}) = T − 1. Moreover, P (σg_{ij}) ≤ P, because the first case of (2) is excluded, as it implies that t_{i}(σ) = −p_{i}(σ) ≤ 0, contrary to our assumption. This completes the induction step.

We are now in the situation that σρ ∈ FSym and loosely speaking we proceed as follows.

(1) We push all irregularities into ray 0, i.e. multiply by ∏ g_{i0} ^{p_{i}(σρ)}.

(2) We push all points back into the ray to which they belong, except for those from ray 0 which we mix into ray 1, say.

(3) We push out of ray 1 separating the points belonging to rays 0 and 1 into ray 0 and any other ray, say ray 2, respectively.

(4) We push the points belonging to ray 1 back from ray 2 into it. These four steps can be achieved by multiplying by an element μ

of length at most 4P̄, such that σρμ is an element which, for each i, permutes an initial segment I_{i} of ray i. It is clear that μ can be chosen so that ∑ |I_{i}| ≤ P̄. Finally, we sort each of these intervals using a recursive procedure, modeled on standard merge sort.

In order to sort the interval I = I_{2} say, we push each of its points out of ray 2 and into either ray 0 if it belongs to the lower half, or to ray 1 if it belongs to the upper half of I. If each of the two halves occurs in the correct order, then we only have to push them back into ray 2 and are done, having used 2|I| generators. If the two halves are not yet sorted, then we use the same “separate the upper and lower halves” approach on each of them recursively in order to sort them. In total this takes at most 2|I| log_{2} |I| steps.

Altogether we have used at most

P + 4P̄ + 2 ∑_{i∈ℤ_{n}} |I_{i}| log_{2} |I_{i}| ≤ 7P log_{2} P

generators to represent the inverse of σ; we used the hypothesis P ≥ 2 in the last inequality. □

We note that because there are many permutations, the fraction of elements which are close to the lower bound goes to zero in much the same way as shown for Thompson’s group V by Birget [1] and its generalization nV by Burillo and Cleary [3].

Lemma 4.2. Let B_{k} be the set of elements of H_{n} with complexity P ≤ k. The fraction of elements of B_{k} which have word length greater than k log k converges exponentially fast to 1.

Proof. In B_{k}, we consider the subset of elements with T = 0 but which do permute at least one of the points at distance k on one of the rays, so T = 0 and P = k. There are at least k! elements of this type. Since the number of elements of word length k in any finitely generated group with d generators is at most (2d)^{k}, the relative growth rates of the factorial and exponential functions give us the result. □

5. Subgroup embeddings

We note that each H_{n} is a subgroup of H_{m} for n < m and that our estimates together with work of Lehnert are enough to give at least quadratic distortion for some of these embeddings.

Theorem 5.1. The group H_{2} is at least quadratically distorted in H_{m} for m ≥ 3.

Proof. We consider the element σ_{n} of H_{2} which has T (σ_{n}) = 0 and transposes (0, k) and (1, k) for all k ≤ n. Then σ_{n} corresponds to the word g_{n} defined in Theorem 8 of [9], where it is shown to have length of the order of n^{2} with respect to the generators of H_{2} in Lemma 10 there, which are exactly the generators for H_{2} given in the introduction. One easily checks that σ_{n} = g_{02} ^{n}g_{12} ^{n}g_{02} ^{−n}g_{12} ^{−n} in H_{3}. Thus a family of words of quadratically growing length in H_{2} has linearly growing length in H_{3}, which proves the theorem. □

A natural, but seemingly difficult, question is whether H_{n} is distorted in H_{m} for 3 ≤ n < m.

6. Some quasi-isometries of H_{n}

Commensurations give rise to quasi-isometries and are often a rich source of examples of quasi-isometries. Here we show that the natural map from the commensurator of H_{n} to the quasi-isometry group of H_{n}, which we denote by QI(H_{n}), is an injection. That is, we show that each commensuration is not within a bounded distance of the identity. That this is an injection also follows from the more general argument of Whyte which appears as Proposition 7.5 in Farb-Mosher [6].

Theorem 6.1. The natural homomorphism from Com(H_{n}) to QI(H_{n}) is an embedding for n ≥ 2.

Proof. We will show that for each non-trivial ϕ ∈ Com(H_{n}) and every N ∈ ℕ we can find a σ ∈ H_{n} such that d(σ, σ^{ϕ}) ≥ N. By Theorem 3.2, we can and will view ϕ as a non-trivial element of N_{p} ⊂ Sym for some even p.

If ϕ eventually translates a ray i non-trivially to a possibly different ray j, then we let σ = ((i, N) (i, N + 1)), a transposition in the translated ray. The image of σ under conjugation by ϕ is the transposition ((j, N + t), (j, N + t′ + 1)), and the distance d(σ, σ^{ϕ}) is the length of σ^{−1}σ^{ϕ}, which is at least N since it moves at least one point at distance N down one of the rays.

If ϕ does not eventually translate a ray but eventually non-trivially permutes ray i with another ray j, then we can show boundedness away from the identity by taking σ = ((j, N) (j, N + 1)). The point (i, N) is fixed by σ but is moved to (i, N + 1) under σ^{ϕ} ensuring that the length of σ^{−1}σ^{ϕ} is at least N.

Finally, if ϕ does not have the preceding two properties, then ϕ is a non-trivial finitary permutation, supported on an interval of size P on ray i and we let σ = g_{i−1} ^{N+P}. The image of σ under ϕ is then a finitary permutation involving points at distance at least N in ray i, giving a lower bound of at least N for d(σ, σ^{ϕ}). □

7. Co-Hopficity

Houghton’s groups are long known to be Hopfian although they are not residually finite, see [4]. In this section we will prove that H_{n} is not co-Hopfian, by exhibiting a map which is injective but not surjective. The map is the following:

\begin{matrix}f: H_{n} ⟶ H_{n} \\ s ↦ f(s) \end{matrix}

defined by: if s(i, n) = (j, m), then:

f(s)(i, 2n − 1) = (j, 2m − 1) and f(s)(i, 2n) = (j, 2m).

It is straightforward to show that f is a homomorphism. It is injective, because if s is not the identity with s(i, n) ≠ (i, n), then f(s)(i, 2n) ≠ (i, 2n). And clearly the map is not surjective, because the permutation always sends adjacent points (i, 2n − 1), (i, 2n) to adjacent points, and a permutation which does not do this cannot be in the image.

Theorem 7.1. Houghton’s groups H_{n} are not co-Hopfian.

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Departament de Matemàtica Aplicada IV, EETAC – UPC, Esteve Terrades 5, 08860 Castelldefels (Barcelona), Spain

E-mail address: burillo@ma4.upc.edu

Department of Mathematics R8133, The City College of New York, Convent Ave & 138th, New York, NY 10031, USA

E-mail address: cleary@sci.ccny.cuny.edu

Mathematics, University of Southampton, Highfield, Southampton, SO17 1BJ, England

E-mail address: A.Martino@soton.ac.uk

Department of Mathematics, National University of Ireland, Galway, University Road, Galway, Ireland

E-mail address: claas.roever@nuigalway.ie