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«Abstract. We prove that even Coxeter groups, whose Coxeter diagrams contain no (4, 4, 2) triangles, are conjugacy separable. In particular, this ...»

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Abstract. We prove that even Coxeter groups, whose Coxeter diagrams contain no

(4, 4, 2) triangles, are conjugacy separable. In particular, this applies to all right-angled

Coxeter groups or word hyperbolic even Coxeter groups. For an arbitrary Coxeter group W, we also study the relationship between Coxeter generating sets that give rise to the same collection of parabolic subgroups. As an application we show that if an automorphism of W preserves the conjugacy class of every sufficiently short element then it is inner. We then derive consequences for the outer automorphism groups of Coxeter groups.

1. Introduction A group G is said to be conjugacy separable if for any two non-conjugate elements x, y ∈ G there is a homomorphism from G to a finite group M such that the images of x and y are not conjugate in M. Conjugacy separability can be restated by saying that each conjugacy class xG := {gxg −1 | g ∈ G} is closed in the profinite topology on G. If G is b residually finite, this also equivalent to the equality xG = xG ∩ G in G for all x ∈ G, where G denotes the profinite completion of G.

Conjugacy separability is a classical notion from Combinatorial Group Theory. Originally it was introduced by Mostowski [Mos66], who suggested the first application of this property by proving that a finitely presented conjugacy separable group has solvable conjugacy problem (see also Malcev’s work [Mal58]). Presently the following classes of groups are known to be conjugacy separable: virtually free groups (Dyer [Dye79]), virtually surface groups (Martino [Mar07]), virtually polycyclic groups (Remeslennikov [Rem69]; Formanek [For76]), finitely presented residually free groups (Chagas and Zalesskii [CZ09]), right angled Artin groups (Minasyan [Min12]), non-uniform arithmetic lattices in SL2 (C) (Chagas and Zalesskii [CZ10]).

While conjugacy separability is a natural amplification of residual finiteness, it is usually much harder to establish. One of the difficulties comes from the fact that, in general, conjugacy separability is not stable under passing to finite index subgroups or overgroups (see [Gor86,CZ09,MM12]). In view of this Chagas and Zalesskii call a group G hereditarily conjugacy separable if every finite index subgroup of G is conjugacy separable. Recent theorems due to Haglund and Wise [HW08, HW10], Wise [Wis] and Agol [Ago] show that many naturally occurring groups possess finite index subgroups that embed into right angled Artin groups as virtual retracts. If G is such a group, then, by the work of the second author [Min12], G contains a hereditarily conjugacy separable subgroup of finite index.

1991 Mathematics Subject Classification. 20F55, 20E26, 20E36.

Key words and phrases. Coxeter group, conjugacy separable, pointwise inner automorphisms.

P-E. C. was partly supported by FNRS grant F.4520.11 and the European Research Council (grant #278469). A. M. was partially supported by the EPSRC grant EP/H032428/1.

2 P.-E. CAPRACE AND A. MINASYAN The goal of the present work is to study conjugacy separability and related properties for Coxeter groups. Recall that a Coxeter group is a group W given by the presentation W = ⟨s1,..., sn ∥ (si sj )mij = 1, for all i, j⟩, (1)

where M := (mij ) is a symmetric n×n matrix, whose entries satisfy the following conditions:

mii = 1 for every i = 1,..., n, mij ∈ N ⊔ {∞} (if mij = ∞, then it is understood that no relation on the product si sj is added in the Coxeter presentation) and mij ≥ 2 whenever 1 ≤ i j ≤ n. The set S = {s1,..., sn } is called the Coxeter generating set for W, M is called the Coxeter matrix and n = |S| is called the rank of W.

Each Coxeter group W is associated with a (free) Coxeter diagram, which is a labeled graph whose vertex set is indexed by the generators {s1,..., sn } such that between vertices corresponding to distinct generators si and sj, there is an edge labeled by mij if and only if mij ̸= ∞. The Coxeter group W is said to be even if all non-diagonal entries in the corresponding Coxeter matrix M are either even integers or ∞. The group W is rightangled if mij ∈ {2, ∞} whenever i ̸= j. Coxeter groups have been an object of intensive study for many years. For background and basic properties of Coxeter groups the reader is referred to [Dav08].

In [NR03] for any given Coxeter group W, Niblo and Reeves construct a CAT(0) cube complex X on which W acts properly by isometries. A combination of Theorem 1.2 from [HW10] and Corollary 2.2 from [Min12] implies that every Coxeter group W, which acts cocompactly on its Niblo-Reeves cube complex, possesses a hereditarily conjugacy separable

subgroup of finite index. Therefore it is natural to ask the following question:

Question 1.1. Is every finitely generated Coxeter group conjugacy separable?

Our first result provides a positive answer to the above question for a large class of even

Coxeter groups:

Theorem 1.2.

Suppose that W is an even Coxeter group of finite rank such that its Coxeter diagram has no (4, 4, 2)-triangles (i.e., no subdiagrams of type B2 ).

Then W is conjugacy separable.

Even Coxeter groups covered by Theorem 1.2 are precisely the ones that act cocompactly on their Niblo-Reeves cube complexes. This follows from a result of the first author and M¨ hlherr [CM05] stating that the action of W on its Niblo-Reeves cubulation is cocompact u if and only if its Coxeter diagram has no irreducible affine subdiagrams of rank at least

3. By the classification of irreducible affine Coxeter groups, in the case when W is even the latter condition is equivalent to the absence of (4, 4, 2)-triangles in the Coxeter diagram of W. In particular, Theorem 1.2 applies if W is right-angled or if W is even and word hyperbolic.

The proof of Theorem 1.2 basically splits into two parts. In the first part we employ a criterion of Chagas and Zalesskii [CZ10] to show that essential elements in W (i.e., elements not contained in any proper parabolic subgroup) are conjugacy distinguished. This relies on the fact that W contains a hereditarily conjugacy separable subgroup of finite index, as discussed above. In the second part, to deal with non-essential elements we introduce a new criterion (Lemma 2.5), which works because standard parabolic subgroups in even Coxeter groups are retracts. In particular we prove that finite order elements are conjugacy distinguished in any even Coxeter group (Proposition 4.1).

Another standard application of conjugacy separability was discovered by Grossman [Gro74], who proved that for a finitely generated conjugacy separable group G, the group of


outer automorphisms Out(G) is residually finite, provided every pointwise inner automorphism of G is inner. Recall that an automorphism α of a group G is called pointwise inner if α(g) is conjugate to g for every g ∈ G. Presently it is unknown whether the outer automorphism group of every finitely generated Coxeter group is residually finite. That some (and conjecturally all) Coxeter groups are conjugacy separable therefore motivates the question whether pointwise inner automorphisms of Coxeter groups are necessarily inner. A positive answer for all finitely generated Coxeter groups is provided by Corollary 1.6 below. This will be deduced from a study of automorphisms that preserve parabolic subgroups. In order to present a precise formulation, we first recall that an automorphism of a Coxeter group is called inner-by-graph if it maps a Coxeter generating set S to a (setwise) conjugate of itself. Such an automorphism is thus a composition of an inner automorphism with a graph automorphism, i.e., an automorphism which stabilises the Coxeter generating set S.

Theorem 1.3.

Let W be a finitely generated Coxeter group with Coxeter generating set S, and let α ∈ Aut(W ) be an automorphism.

Then α is inner-by-graph if and only if it satisfies the following two conditions:

(1) α maps every parabolic subgroup to a parabolic subgroup.

(2) For all s, t ∈ S, such that st has finite order in W, there is a pair s′, t′ ∈ S such that α(st) is conjugate to s′ t′.

Theorem 1.3 will follow from Proposition 6.

5 and Theorem 7.1 below. The condition (2) in Theorem 1.3 can be interpreted geometrically: it means that the reflections s and t are mapped by α to a pair of reflections such that the angle between their fixed walls is preserved. In the terminology recalled in Section 6 below, we say that the Coxeter generating sets S and α(S) are angle-compatible (cf. Lemma 7.2). For a thorough study of the relation of angle-compatibility, we refer to [MM08].

It is easy to see that condition (2) is necessary for α to be inner-by-graph: examples illustrating that matter of fact may be found among finite dihedral groups. It turns out, however, that is W is crystallographic, i.e., if mij ∈ {2, 3, 4, 6, ∞} for all i ̸= j, then condition (2) is automatically satisfied. In particular we obtain Corollary 1.4. Let W be a finitely generated crystallographic Coxeter group. Then an automorphism α ∈ Aut(W ) is inner-by-graph if and only if α maps every parabolic subgroup to a parabolic subgroup.

We shall also see that an automorphism of W preserving the conjugacy class of every element of small word length (with respect to S) also satisfies the conditions of Theorem 1.3.

In fact, in such a case one can even exclude graph automorphisms, thereby yielding the following corollary.

Corollary 1.5. Let W be a finitely generated Coxeter group with Coxeter generating set S, and let α ∈ Aut(W ) be an automorphism. Suppose that α(w) is conjugate to w for all elements w that can be written as products of pairwise distinct generators (in particular the word length of such elements is bounded above by |S|).

Then α is inner.

The following consequence of Corollary 1.5 is immediate:

Corollary 1.6. Every pointwise inner automorphism of a finitely generated Coxeter group is inner.

4 P.-E. CAPRACE AND A. MINASYAN Combining Corollary 1.6, Theorem 1.2 together with the theorem of Grossman [Gro74] mentioned above we obtain the following.

Corollary 1.7. Assume that W is a finitely generated even Coxeter group whose Coxeter diagram contains no (4, 4, 2)-triangles. Then Out(W ) is residually finite.

–  –  –

Proposition 2.2 was used by Chagas and Zalesskii to show that certain torsion-free extensions of hereditarily conjugacy separable groups are conjugacy separable (see [CZ10]).

However, in order to deal with torsion we need to find a different criterion.

Let G be a group and let A be a subgroup of G. Recall that an endomorphism ρA : G → G is called a retraction of G onto A if ρA (G) = A and ρA (h) = h for every h ∈ A. In this case A is said to be a retract of G. Note that ρA ◦ ρA = ρA.

Assume that A and B are two retracts of a group G and ρA, ρB ∈ End(G) are the corresponding retractions. We will say ρA commutes with ρB if they commute as elements of the monoid of endomorphisms End(G), i.e., if ρA (ρB (g)) = ρB (ρA (g)) for all g ∈ G.

Remark 2.4 (Rem.

4.2 in [Min12]). If the retractions ρA and ρB commute then ρA (B) = A ∩ B = ρB (A) and the endomorphism ρA∩B := ρA ◦ ρB = ρB ◦ ρA is a retraction of G onto A ∩ B.

Indeed, obviously the restriction of ρA∩B to A ∩ B is the identity map. And ρA∩B (G) ⊆ ρA (G) ∩ ρB (G) = A ∩ B, hence ρA∩B (G) = A ∩ B. Consequently ρA (B) = ρA (ρB (G)) = ρA∩B (G) = A ∩ B. Similarly, ρB (A) = A ∩ B.

6 P.-E. CAPRACE AND A. MINASYAN Lemma 2.5. Suppose that A, B G are retracts of G such that the corresponding retractions ρA, ρB ∈ End(G) commute. Then for arbitrary elements x ∈ A and y ∈ B, x is

conjugate to y in G if and only if the following three conditions hold:

(1) ρA (y) ∈ xA in A;

(2) ρB (x) ∈ y B in B;

(3) ρA∩B (y) ∈ ρA∩B (x)A∩B in A ∩ B.

Proof. Suppose that y = g −1 xg for some g ∈ G. Applying ρA to both sides of this equality we achieve ρA (y) = ρA (g)−1 xρA (g), thus ρA (y) ∈ xA. Similarly, ρB (x) ∈ y B. Finally, (3) follows after applying ρA∩B to both sides of the above equality.

Assume, now, that the conditions (1)–(3) hold. Note that ρA∩B (x) = ρB (ρA (x)) = ρB (x) as x ∈ A; similarly, ρA∩B (y) = ρA (y). Then x is conjugate to ρA (y) = ρA∩B (y), which is conjugate to ρA∩B (x) = ρB (x), which is conjugate to y in G. Since conjugacy is a transitive relation we can conclude that y ∈ xG.

3. Parabolic subgroups and parabolic closures in Coxeter groups In this section we collect some of the basic facts about parabolic subgroups of Coxeter group that will be used in the rest of the paper.

Let W be a Coxeter group with a fixed finite Coxeter generating set S. In this section we will remind some terminology and basic properties of W and its parabolic subgroups. A reflection of W is an element conjugate to some s ∈ S. Given J ⊆ S, we set WJ = ⟨J⟩.

A subgroup of the form WJ for some J ⊆ S is called a standard parabolic subgroup of W. It is a standard fact that WJ is itself a Coxeter group with Coxeter generating set J.

A subgroup P is called parabolic if it is conjugate to some standard parabolic subgroup WJ. The rank rank(P ) of that parabolic subgroup is the cardinality of J.

The following basic property of parabolic subgroups is crucial.

Lemma 3.1.

Let P, Q be two parabolic subgroups of a Coxeter group W. Then P ∩ Q is a parabolic subgroup with respect to the Coxeter group Q (and the natural Coxeter generating set of Q coming from S).

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