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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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Proof. We recall that the Cayley graph of a Coxeter group may be viewed as a chamber system; this fact, as well as a basic introduction to chamber systems, can be found in [Wei03]. We recall that parabolic subgroups in a Coxeter group are exactly the stabilisers of the residues. Given parabolic subgroups P Q, let RP and RQ be the residues whose ′ stabilisers are precisely P and Q. Then the combinatorial projection RP = projRQ (RP ) of RP on RQ is a residue stabilised by P ∩ Q (see [Tit74, Prop. 2.29]). Moreover, properties of ′ the combinatorial projection imply that if a reflection stabilises RP, then it also stabilises ′ also stabilises R, since it is generated by reflections.

RP. It follows that the stabiliser of RP P ′ ′ Finally, since RP is contained in RQ, the stabiliser of RP is also contained in Q This shows ′ that the stabiliser of RP equals P ∩ Q. Thus P ∩ Q is a parabolic subgroup in the Coxeter group Q, as claimed.

Lemma 3.1 implies that any intersection of parabolic subgroups is a parabolic subgroup.

In particular any subset H of W is contained in a unique minimal parabolic subgroup, called the parabolic closure of H. Moreover, if P, Q are parabolic subgroups such that P is properly contained in Q, then the rank of P is strictly smaller than the rank of Q (for the definition and basic properties of parabolic closures, see [Kra09, §2.1]).

Lemma 3.2.

Let H W be a finite subgroup generated by n reflections. Then Pc(H) is a finite parabolic subgroup of rank ≤ n.


Proof. The fact that Pc(H) is finite is well-known, see [Bou68, Ch. V, §4, Exercice 2.d].

Now it suffices to show that in a finite Coxeter group W, a reflection subgroup generated by n reflections is contained in a parabolic subgroup of rank n. Let Σ be the geometric realization of the Coxeter complex of W (see [Tit74, Ch. 2] or [AB08, Ch. 3] for Coxeter complexes). Each reflection fixes pointwise a hyperplane of the sphere Σ. Thus H fixes a subcomplex Σ′ of codimension d ≤ n. Let σ ⊂ Σ′ be a simplex of codimension d. Then H is contained in P = StabW (σ) and P is a parabolic subgroup of rank d ≤ n.

Let J ⊆ S. We set J ⊥ = {s ∈ S \J | sj = js for all j ∈ J}. The set J is called spherical if WJ is finite. The set J ⊆ S is called irreducible if for every non-empty subset I J, we have J ̸⊂ I ∪ I ⊥ ; equivalently the parabolic subgroup WJ does not split as a direct product of proper parabolic subgroups. It is a fact that if an infinite Coxeter group W admits an irreducible Coxeter generating set S, then any other Coxeter generating set of W is also irreducible. (If W is finite, this is, however, not the case, since a dihedral group of order 12 is the direct product of a group of order 2 and a dihedral group of order 6.) Thus, in that case, it makes sense to say that W itself is irreducible.

Lemma 3.3.

Let J ⊆ S be irreducible and non-spherical.

(i) NW (WJ ) = WJ∪J ⊥ and CW (WJ ) = WJ ⊥.

(ii) If wJw−1 ⊂ S for some w ∈ W, then wJw−1 = J.

(iii) If J ⊥ = ∅, then every parabolic subgroup of W containing WJ is standard.

Proof. For (i) and (ii), see [Deo82] or [Kra09, §3.1]. Assertion (iii) is well known to the experts and can be deduced from (i). By lack of an appropriate reference, we provide a proof. To this end, we view the Cayley graph X of W with respect to S as a chamber system (see [AB08, §5.2] for the definition of chamber systems and the associated terminology).

For each I ⊆ S, the parabolic subgroup WI is the stabiliser in W of the I-residue of X containing the base chamber 1, which is denoted by ResI (1).

Let now R and R′ be two residues whose stabiliser in W is WJ. For every wall W crossed by a minimal gallery joining a chamber in R to its projection to R′, the wall W does not cross R′ (by properties of the projection) and, hence, the associated reflection rW does not stabilise R′. Since R and R′ have the same stabiliser, we infer that W does not cross R either.

This proves that every wall crossed by a gallery joining a chamber in R to its projection to R′, separates R from R′. It follows that such a wall W is contained in a bounded neighborhood of R. Therefore the reflection rW commutes with WJ by [CM12, Lemma 2.20]. From (i) and the hypothesis that J ⊥ is empty, we infer that there is no wall separating R from R′.

In other words ResJ (1) is the unique residue in X whose stabiliser is WJ.

Let now P be a parabolic subgroup containing WJ. Then P is the stabiliser of some residue R. Since WJ P it follows that R contains a residue whose stabiliser is WJ.

Thus R contains ResJ (1) by what we have just proved. It follows that R is of the form R = ResJ∪J ′ for some J ′ ⊆ S \ J which implies that P is indeed standard.

Lemma 3.4.

Let J = {s1,..., sn } ⊆ S and denote by w = s1 s2... sn the product of all elements of J (ordered arbitrarily). Then Pc(w) = WJ.

Proof. See Theorem 3.4 in [Par07] or Corollary 4.3 in [CF10].

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is closed in the subspace topology on A′, induced by the profinite topology of K. In view of ′ Remark 2.1 we see that aA is separable in A′. Thus any finite index subgroup A′ A ∩ H is conjugacy separable, i.e., A ∩ H is hereditarily conjugacy separable.

Lemma 4.5.

Any amenable subgroup of a finitely generated Coxeter group is closed in the profinite topology.

Proof. Coxeter groups are CAT(0) groups by [Dav08, Th. 12.3.3]. Therefore every amenable subgroup is virtually abelian by [AB98, Cor. B].

By a theorem of Haglund and Wise [HW10, Cor. 1.3], any finitely generated Coxeter group W contains a finite index subgroup G such that G is a subgroup of some right angled Coxeter group R of finite rank. The standard geometric representation of R (see [Hum90, 5.3]) is a faithful representation ([Hum90, Cor. 5.4]) by matrices with integer coefficients. It follows that G is a finitely generated subgroup of GLn (Z) for some n ∈ N.

Segal proved (see [Seg83, Thm. 5, p. 61]) that every solvable subgroup of GLn (Z) is closed in the profinite topology of that group. Hence every solvable subgroup of G is separable in G (by Remark 2.1).

So, let L be a virtually solvable subgroup of W. Since |W : G| ∞ we can find a solvable ⊔ L ∩ G and f1,..., fk ∈ L such that L = k fi M. By the above, M is subgroup M i=1 separable in G, therefore, according to Remark 2.1, it is also separable in W. Consequently, L is closed in PT (W ) as a finite union of closed sets.

We will now apply the Chagas-Zalesskii criterion [CZ10] to obtain Lemma 4.6. Let W be an infinite non-affine irreducible Coxeter group of finite rank. If W has a finite index hereditarily conjugacy separable subgroup H then every essential element in W is conjugacy distinguished.

Proof. Evidently we can assume that H is normal in W. Consider any essential element x ∈ W. Set m := |W : H| ∈ N, then xm is also essential in W by Lemma 4.2. Therefore, according to Lemma 4.3, the centralizer CW (xm ) is virtually cyclic and hence it is conjugacy separable (cf. [Rem69, For76]). Also, every subgroup of CW (xm ) is virtually cyclic, and so it is separable in W by Lemma 4.5. Therefore we can apply Proposition 2.2 to conclude that xW is separable in W.

The proof of the next statement combines the criteria from Proposition 2.2 and Lemma 2.5.

Proposition 4.7. Suppose that W is an even Coxeter group of finite rank that contains a finite index normal subgroup H ▹ W such that H is hereditarily conjugacy separable. Then W is conjugacy separable.

Proof. The proof will proceed by induction on the rank rank(W ) = |S|, where S is a fixed Coxeter generating set of W. If rank(W ) ≤ 1 then W is finite and the claim trivially holds. So suppose that rank(W ) 1 and the claim has already been established for all even Coxeter groups of rank less than rank(W ). If W is finite then there is nothing to prove; if W is affine, then it is virtually abelian and so it is conjugacy separable (as any virtually polycyclic group – see [Rem69, For76]).

If W is not irreducible, then W = WI × WJ for some I, J S such that S = I ⊔ J.

Note that WI is an even Coxeter group with rank(WI ) = |I| |S| = rank(W ) and WI ∩ H 10 P.-E. CAPRACE AND A. MINASYAN

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Proof of Theorem 1.2. By Corollary 1.5 from [CM05], any Coxeter group whose Coxeter diagram does not contain irreducible affine subdiagrams of rank at least 3 acts cocompactly on the associated Niblo–Reeves cube complex (see [NR03]). Since the only even irreducible affine Coxeter diagram of rank ≥ 3 is B2 (according to the classification of all irreducible affine Coxeter groups – see, for example, [Dav08, Appendix C]), our assumptions imply that W acts cocompactly on its Niblo–Reeves cubing.

As discussed in the introduction, the results of Haglund and Wise from [HW08, HW10] combined with the main theorem of [Min12] imply that every Coxeter group, whose action on the associated Niblo–Reeves cube complex is cocompact, has a hereditarily conjugacy separable subgroup of finite index. Therefore, W satisfies all the assumptions of Proposition 4.7, allowing us to conclude that it is conjugacy separable.

5. Reflection subgroups of Coxeter groups A reflection subgroup of W is defined as a subgroup of W generated by reflections.

For example each parabolic subgroup is a reflection subgroup. It is a general fact that a reflection subgroup is itself a Coxeter group. We shall need the following more precise version of this fact.

Proposition 5.1. Let G W be a reflection subgroup.

(i) There is a set of reflections R ⊂ G such that (G, R) is a Coxeter system.

(ii) Let Γ(W,S) (resp. Γ(G,R) ) be the Cayley graph of (W, S) (resp. (G, R)). Let Γ be the quotient graph of Γ(W,S) obtained by collapsing each edge stabilised by a reflection which does not belong to G. Then Γ is G-equivariantly isomorphic to Γ(G,R).

Proof. See [Deo89] or [Dye90].


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6. Automorphisms preserving parabolic subgroups up to conjugacy Let W be a finitely generated Coxeter group. Two Coxeter generating sets S1, S2 for W are called reflection-compatible if each element of S1 is conjugate to an element of S2. They are called angle-compatible if they are reflection-compatible and if, moreover, for each spherical pair {s, t} ⊆ S1, there is w ∈ W such that {wsw−1, wtw−1 } ⊆ S2.

Furthermore, we say that S1 and S2 are parabolic-compatible if for every J1 ⊆ S1, there is some J2 ⊆ S2 such that the subgroup WJ1 is conjugate to WJ2.

It is important to remark that reflection-compatibility, angle-compatibility and paraboliccompatibility are equivalence relations on the collection of all Coxeter generating sets. For the first two relations, see [CP10, Appendix A]; for parabolic-compatibility, this follows from Lemma 6.4 below.

12 P.-E. CAPRACE AND A. MINASYAN The following basic observation is useful.

Lemma 6.1.

Let W be a finitely generated Coxeter group. Any two Coxeter generating sets which are reflection-compatible have the same cardinality.

Proof. Follows from basic considerations using root systems. The desired statement boils down to the property that any two bases of a vector space have the same cardinality.

Remark that two Coxeter generating sets that are not reflection-compatible need not have the same cardinality. For example, the dihedral group of order 12 is isomorphic to the direct product of the dihedral group of order 6 with the cyclic group of order 2.

Clearly, the relation of parabolic-compatibility is much stronger than reflection-compatibility among Coxeter generating sets for W. For example, if W is a free Coxeter group, i.e., a free product of groups of order 2, then any two Coxeter generating sets are reflection compatible (because any involution in W is a reflection in that case), but if the rank of W is at least 3, it is easy to find automorphisms that do not map every parabolic subgroup to a parabolic subgroup.

The following lemma shows however that reflection-compatibility is sufficient to ensure the compatibility of all spherical parabolic subgroups.

Lemma 6.2.

Let S, S ′ be reflection-compatible Coxeter generating sets for a Coxeter group W. Then for each spherical subset J ⊆ S, there is a subset J ′ ⊆ S ′ with |J| = |J ′ | such that WJ and WJ ′ are conjugate.

Proof. Let n = |J| and P = Pc(WJ ) be the parabolic closure of WJ with respect to the Coxeter generating set S ′. Then P is a finite parabolic subgroup of rank k ≤ n with respect to S ′, by Lemma 3.2. The lemma also implies that the parabolic closure Q of P with respect to S is a parabolic subgroup of rank k ′ ≤ k ≤ n with respect to S. Since WJ P, we have WJ Q. Since WJ is of rank n and Q is of rank k ′ ≤ n, it follows that WJ = Q and k ′ = n.

In particular WJ = P = Q and k = k ′ = n so that WJ is a finite parabolic of rank n with respect to S ′, as desired.

We shall also need the following technical fact, showing that the various notions of compatibility are appropriately inherited by parabolic subgroups.

Lemma 6.3.

Let W be a finitely generated Coxeter group, and S1, S2 be two Coxeter generating sets. Let J1 ⊆ S1 and J2 ⊆ S2 be such that WJ1 = WJ2.

If S1 and S2 are reflection-compatible (resp. angle-compatible, parabolic-compatible), then so are J1 and J2 as Coxeter generating sets for the Coxeter subgroup WJ1 = WJ2.

Proof. We start with the following observation, which is a special case of Lemma 3.1: if P, Q are parabolic subgroups of a Coxeter group W and if P is contained in Q, then P is also parabolic as a subgroup of the Coxeter group Q.

The above observation is already enough to draw the desired conclusion for reflectioncompatibility and parabolic-compatibility.

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