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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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Assume, now, that S1 and S2 are angle-compatible. This implies that in the Cayley graph of (W, S2 ), viewed as a chamber system, every spherical pair {s, t} ⊆ S1 fixes two walls that contain two panels σ, τ of a common chamber, say c. Given any residue R stabilised by P = ⟨s, t⟩, consider the combinatorial projection of c, σ and τ onto R, and call them c′, σ ′ and τ ′. By properties of the projection (see [Tit74, 2.30–2.32]), σ ′ and τ ′ must be panels stabilised by s and t respectively; moreover, they are both panels of the chamber c′.

We now fix a residue R0, whose stabiliser is WJ1 = WJ2. If the pair {s, t} is contained in J1, then, by Lemma 3.2, we can find a rank two residue R stabilised by P within R0. By the

ON CONJUGACY SEPARABILITY OF SOME COXETER GROUPS 13

–  –  –

The goal of this section is to establish the following fact, which will later be used to prove Theorem 1.3 from the Introduction.

Proposition 6.5. Let W be a finitely generated Coxeter group, and S1, S2 be two Coxeter generating sets.

If S1, S2 are angle-compatible and parabolic-compatible, then there is some inner automorphism α ∈ Inn(W ) such that α(S1 ) = S2.

The condition that S1 and S2 are parabolic-compatible is not sufficient on its own to guarantee that they are conjugate; examples illustrating this matter of fact may be found amongst finite dihedral groups.

A subset of a Coxeter generating set is called 2-spherical if every pair of elements in it is spherical. We shall need the following elementary fact.

Lemma 6.6.

Let W be a Coxeter group with Coxeter generating set S. Let J ⊆ S be a subset that is irreducible, but not 2-spherical. If |J| 2, then there is some s ∈ J such that J \ {s} is still irreducible and non-2-spherical.

Proof. Start with a pair of elements I0 ⊂ J that generates an infinite dihedral group. Since J is irreducible, the pair I0 must be contained in a triple I1 ⊂ J which is still an irreducible subset. Proceeding inductively, we construct a chain I0 I1... such that |In | = n + 2 and each Ii is irreducible and not 2-spherical. The result follows.

–  –  –

which must a fortiori be the common vertex between the edges fixed by s1 and s2. Therefore s0 = s′, and we are done.

If {s0, s1 } is spherical and {s0, s2 } is not, then W is the free product of a finite dihedral group ⟨s0, s1 ⟩ and a cyclic group of order 2 generated by s2. This decomposition of W as a free product must also be visible with respect to the Coxeter generating set S2. Indeed, the pair {s′, s2 } is non-spherical (because ⟨s2 ⟩ is a free factor of W of order 2). Therefore the pair {s′, s1 } must be spherical, by angle-compatibility of S1 and S2.

We next remark that ⟨s0, s1 ⟩ is the unique maximal finite subgroup of W containing s1.

Since ⟨s′, s1 ⟩ is such a finite subgroup, we must have s′ ∈ ⟨s0, s1 ⟩. Now, the property that s′ s2 is a translation of length 2 in the Cayley graph of (W, S1 ) forces s′ = s0, as desired.

If {s0, s1 } and {s0, s2 } are both spherical, then W splits as the amalgamated product W = ⟨s1, s0 ⟩ ∗⟨s0 ⟩ ⟨s0, s2 ⟩. We claim that W contains a unique non-trivial element x = s0 such that both of the subgroups ⟨x, s1 ⟩ and ⟨x, s2 ⟩ are finite.

Since the pairs {s1, s′ } and {s2, s′ } are both spherical (otherwise W would have ⟨s1 ⟩ or ⟨s2 ⟩ as a free factor of order two, which is impossible since {s0, s1 } and {s0, s2 } are both spherical), that claim readily implies that s0 = s′, which concludes the proof in the special case at hand.

The claim can be established as follows. Since W is an amalgamated product, it acts on the associated Bass-Serre tree T. Suppose that x ∈ W is an element such that both of the subgroups ⟨x, s1 ⟩ and ⟨x, s2 ⟩ are finite. A finite group acting on a tree always fixes some vertex (see [Ser80, Example I.6.3.1]), hence there are vertices u1, u2 of T such that u1 ∈ Fix(x) ∩ Fix(s1 ) and u2 ∈ Fix(x) ∩ Fix(s2 ), where Fix(x) denotes the set of vertices of T fixed by x. Note that Fix(s1 ) and Fix(s2 ) are two convex subsets of T with empty intersection, because vertex stabilisers (for the action of W on T ) are finite and the pair {s1, s2 } is not spherical by the assumptions. Therefore there is a unique edge e with e− ∈ Fix(s1 ) and e+ ∈ Fix(s2 ). The stabiliser of e in W is the subgroup ⟨s0 ⟩ and any arc in T connecting a vertex of Fix(s1 ) with a vertex Fix(s2 ) must pass through e. Since x fixes one of such arcs [u1, u2 ], we deduce that x fixes e. As ⟨s0 ⟩ contains only one non-trivial element, we can conclude that x = s0, thereby proving the claim.

Assume now that S ′ has more than two elements. By Lemma 6.6, there is some s1 ∈ S ′ such that S ′′ = S1 \ {s1 } is still irreducible and not 2-spherical. Since S ′ is irreducible, it follows that (S ′′ )⊥ ⊆ {s0 }, where for a subset J ⊆ S, J ⊥ denotes the set of those s ∈ S \ J commuting with J in W.

If (S ′′ )⊥ = {s0 }, then the centraliser of S ′′ in W is ⟨s0 ⟩ by Lemma 3.3(i). Applying the same lemma with respect to the Coxeter generating set S2 then yields that the centraliser of S ′′ is ⟨s′ ⟩. Therefore s0 = s′ and we are done in this case.





If (S ′′ )⊥ = ∅, then WS ′ and WS ′′ ∪{s0 } are the only two proper parabolic subgroups of W (with respect to the Coxeter generating set S1 ) containing WS ′′ properly, by Lemma 3.3(iii).

Since S1 and S2 are parabolic-compatible, we infer that WS ′′ ∪{s0 } = WS ′′ ∪{s′ }.

By Lemma 6.3, the sets S ′′ ∪ {s0 } and S ′′ ∪ {s′ } are angle- and parabolic-compatible.

Thus by induction there is some w ∈ WS ′′ ∪{s0 } such that wS ′′ w−1 ∪ {ws′ w−1 } = S ′′ ∪ {s0 }.

Since S ′′ is irreducible non-spherical, it follows from Lemma 3.3(ii) that wS ′′ w−1 = S ′′, which implies that ws′ w−1 = s0. Moreover, since w normalizes WS ′′ and since (S ′′ )⊥ = ∅, we infer from Lemma 3.3(i) that w must be trivial. Hence s0 = s′ and we are done.

Corollary 6.7. Let W be a finitely generated Coxeter group with Coxeter generating set S and let α ∈ Aut(W ). Then α is inner-by-graph if and only if α(S) is angle-compatible with S, and α maps every parabolic subgroup to a parabolic subgroup.

16 P.-E. CAPRACE AND A. MINASYAN Proof. The necessity is obvious. And the sufficiency follows by applying Proposition 6.5 to the Coxeter generating sets S1 = S and S2 = α(S).

We also deduce the following criterion ensuring that an automorphism is inner.

Corollary 6.8. Let W be a finitely generated Coxeter group with Coxeter generating set S and let α ∈ Aut(W ). Then α is inner if and only if α(S) is angle-compatible with S, and α maps every parabolic subgroup to a conjugate of itself.

Proof. The necessity is trivial. For the sufficiency, suppose that S and α(S) are two Coxeter generating sets which are reflection-compatible, angle-compatible and parabolic-compatible.

Proposition 6.5 ensures that, after replacing α by some appropriate element from the coset αInn(W ), we may assume that α(S) = S. It then follows from Lemma 6.9 below that α is inner.

Lemma 6.9.

Let W be a finitely generated Coxeter group with Coxeter generating set S and α ∈ Aut(W ) an automorphism such that α(S) = S. If α maps every parabolic subgroup of W to a conjugate parabolic subgroup, then α is inner.

Proof. We first notice that α preserves each irreducible component of S. There is thus no loss of generality in assuming that S is irreducible.

Let J ⊆ S be a subset. We claim that if J is irreducible and non-spherical (resp. maximal spherical), then α(J) = J. Indeed WJ is conjugate to α(WJ ) = Wα(J) by hypothesis. By [Deo82], two irreducible non-spherical (resp. maximal spherical) subsets of S are conjugate if and only if they coincide. Thus J = α(J) and the claim stands proven.

Assume now that S is non-spherical. Let J ⊆ S be irreducible non-spherical and minimal with these properties.

Then α(J) = J by the claim above. Moreover, since S is irreducible, we can order the elements of S \ J, say S \ J = {t1,..., tk }, so that J ∪ {t1,..., ti } is irreducible (and nonspherical) for all i. Applying the claim to each of these sets, we deduce that α(ti ) = ti for all i k.

Since J is minimal non-spherical, it follows that for each s ∈ J, the subset Js = J \ {s} is contained in some maximal spherical subset of S not containing s. Applying the claim to such a maximal spherical subset, we infer that α(Js ) = Js. Hence α(s) = s. Thus α acts trivially on S and we are done in this case.

Assume finally that S is spherical. The types of the irreducible finite Coxeter groups admitting a non-trivial graph automorphisms are: An (n 1), Dn (n 3), E6, F4 and the dihedral groups I2 (n). For types An (with n arbitrary), Dn (with n odd), E6 and I2 (n) (with n odd), the unique non-trivial graph automorphism is inner and realized by the longest element. For type F4 and I2 (n) with n even, the unique non-trivial graph automorphism swaps the two conjugacy classes of reflections. Therefore it does not preserve the conjugacy classes of parabolic subgroups of rank one. Finally, for W of type Dn with n even and α ∈ Aut(W ) a non-trivial graph automorphism, we let J ⊂ S be one of the two maximal irreducible proper subsets which is not α-invariant. Since WJ is conjugate to Wα(J), [Deo82] implies that J = α(J), a contradiction.

–  –  –

Theorem 7.1.

Let S and S ′ be Coxeter generating sets of a finitely generated Coxeter group W.

Then there is an inner automorphism α ∈ Inn(W ) such that α(S) = S ′ if and only if the

following two conditions are satisfied:

(1) For each J ⊆ S, there is J ′ ⊆ S ′ such that WJ and WJ ′ are conjugate.

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

Lemma 7.2.

Let S, S ′ be reflection-compatible Coxeter generating sets for a Coxeter group W. Suppose that for each spherical pair {s, t} ⊆ S there is a spherical pair {s′, t′ } ⊆ S ′ such that st is conjugate to s′ t′.

Then S and S ′ are angle-compatible.

Proof. Let {s, t} ⊆ S be a spherical pair. After replacing S ′ with a conjugate, we may assume that st = s′ t′. By Lemma 5.2, the parabolic closure of st = s′ t′ with respect to S (resp. S ′ ) is the group W{s,t} (resp. W{s′,t′ } ). On the other hand, Lemma 6.2 ensures that W{s,t} is parabolic with respect to S ′ and W{s′,t′ } is parabolic with respect to S. It follows that W{s,t} = W{s′,t′ }.

It is easy to verify that any Coxeter generating pair s′, t′ of the finite dihedral group W{s,t} such that the rotations st and s′ t′ coincide, must be setwise conjugate to {s, t} within the group W{s,t}. Therefore S and S ′ are angle-compatible, as desired.

Proof of Theorem 7.1. That conditions (1) and (2) are necessary is clear. Assume that (1) and (2) hold. Thus S and S ′ are parabolic-compatible by (1). They are also anglecompatible by (2), in view of Lemma 7.2. Hence the conclusion follows from Proposition 6.5.

Proof of Theorem 1.3. The claim is immediate from Theorem 7.1 applied to the Coxeter generating sets S and S ′ = α(S).

Proof of Corollary 1.4. In view of Theorem 1.3, it suffices to show that if α ∈ Aut(W ) satisfies (1) from that theorem, then it also satisfies (2). Given a spherical pair {s, t} ⊆ S, we know that α(W{s,t} ) is a spherical parabolic of rank two (by (1) and Lemma 6.2). Thus, after replacing α with some automorphism from the same coset αInn(W ), we can suppose that α(W{s,t} ) = W{s′,t′ } for some {s′, t′ } ⊆ S. Since S is reflection-compatible with α(S), by the assumptions, Lemma 6.3 implies that the generating pairs {s′, t′ } and {α(s), α(t)} are reflection-compatible in W{s′,t′ }. It remains to observe that in a finite dihedral group of order 4, 6, 8 or 12, any two reflection-compatible Coxeter generating pairs are automatically angle-compatible. Thus the pairs {s, t} and {α(s), α(t)} are setwise conjugate, so that S and α(S) are angle-compatible. In particular condition (2) holds, as desired.

Proof of Corollary 1.5. Let S ′ = α(S). Clearly S ′ is a Coxeter generating set which is reflection-compatible with S. Moreover S and S ′ are angle-compatible by Lemma 7.2.

We claim that α maps every parabolic subgroup to a conjugate of itself. Indeed, let J ⊆ S and let xJ be the product of the elements of J ordered arbitrarily. By hypothesis α(xJ ) is conjugate to xJ. Since α maps each reflection to a reflection, it maps a reflection subgroup to a reflection subgroup, and it follows therefore from Lemma 5.2 that α(WJ ) is conjugate to WJ. Therefore α maps every standard parabolic subgroup to some conjugate of itself.

Thus all the hypotheses of Corollary 6.8 are satisfied, thereby yielding the claim.

18 P.-E. CAPRACE AND A. MINASYAN Acknowledgements. We are grateful to Piotr Przytycki for a careful reading of an earlier version of this note.

–  –  –

[Mar07] Armando Martino, A proof that all Seifert 3-manifold groups and all virtual surface groups are conjugacy separable, J. Algebra 313 (2007), no. 2, 773–781.

[MM12] Armando Martino and Ashot Minasyan, Conjugacy in normal subgroups of hyperbolic groups, Forum Math. 24 (2012), no. 5, 889–909.

[Min12] Ashot Minasyan, Hereditary conjugacy separability of right angled Artin groups and its applications, Groups Geom. Dyn. 6 (2012), no. 2, 335–388.

[Mos66] A. Wlodzimierz Mostowski, On the decidability of some problems in special classes of groups, Fund.

Math. 59 (1966), 123–135.

[NR03] Graham A. Niblo and Lawrence D. Reeves, Coxeter groups act on CAT(0) cube complexes, J.

Group Theory 6 (2003), no. 3, 399–413.

[Rem69] Vladimir N. Remeslennikov, Conjugacy in polycyclic groups, Algebra i Logika 8 (1969), 712–725 (Russian).

[Par07] Luis Paris, Irreducible Coxeter groups, Internat. J. Algebra Comput. 17 (2007), no. 3, 427–447.



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