Rigidity of skew-angled Coxeter groups

Abstract

A Coxeter system is called skew-angled if its Coxeter matrix contains no entry equal to 2. In this paper we prove rigidity results for skew-angled Coxeter groups. As a consequence of our results we obtain that skew-angled Coxeter groups are rigid up to diagram twisting.3. If S has at least 3 elements and if GðW ; SÞ is edge-connected, then there exists w A W such that S w ¼ S.Reformulating statement (1) of the main theorem in the language of [3] we get the following corollary. It implies that Conjecture 8.1 in [3] holds in the skew-angled case.Corollary A. Skew-angled Coxeter systems are reflection-rigid up to diagram twisting.Remark 1. Let ðW ; SÞ be a Coxeter system and let s; t A S be two reflections corresponding to the vertices which are on a bridge of GðW ; SÞ. If there is a non-trivial reflection-preserving outer automorphism a of hs; ti (like for instance in the case where st has order 5), then it has an extension to a reflection-preserving automorphism b of W and bðSÞ is not twist equivalent to S because twistings are 'anglepreserving'.If ðW 1 ; S 1 Þ and ðW 2 ; S 2 Þ are both skew-angled Coxeter systems then any isomorphism f : W 1 ! W 2 maps reflections onto reflections since the parabolic dihedral subgroups are the maximal finite subgroups and since any automorphism of a dihedral group D 2n with n d 3 maps reflections onto reflections. The theorem therefore gives a solution to the isomorphism problem for the class of skew-angled Coxeter groups. This can be rephrased as follows:Corollary B. Given two fundamental sets S; S 0 in a Coxeter group W such that ðW ; SÞ and ðW ; S 0 Þ are skew-angled, then GðW ; SÞ and GðW ; S 0 Þ are twist equivalent.As there are no spikes in an edge-connected graph, Part 3 of the main theorem and the previous theorem have the following consequence:Corollary C. Skew-angled Coxeter systems whose diagram has no spike which is labelled by twice an odd number are rigid up to diagram twisting (in the sense of [3]);

Automorphisms and abstract commensurators of 2–dimensional Artin groups

“…We remark that the classification up to isomorphism is already contained in the work of Mühlherr and Weidmann [18] on reflection rigidity and reflection independance in large type (what they call "skew-angled") Coxeter groups. Also, the automorphism groups have already been determined in many of the cases covered here (and some besides) by Bahls [2].…”

“…Recently, Mühlherr and Weidmann [18] have proved results on reflection rigidity and reflection independance in the wider class of large type (LT) Coxeter groups which give the same solution to the classification problem as given by Theorem 4 above. We note that Bahls [3] has also obtained a similar classification for those Coxeter groups having 2-dimensional Davis complex (equivalently, those associated to 2-dimensional Artin groups).…”

On the Isomorphism Problem for Coxeter Groups and Related Topics

“…Indeed, it is crucial in most of the preceding arguments for finitely-generated case that a maximal finite subgroup containing a given element always exists in these cases; however, this property is not assured in general case. (Note that the preceding arguments still work in certain cases; see [MW02] and [Bah05] for instance. )…”

“…In [BMMN02], the four authors in that paper proposed the following important conjecture. For example, this conjecture is proved for finitely generated skew-angled Coxeter groups [MW02]. Note that the finiteness of the Coxeter graphs is not assumed in the original conjecture, but a counterexample exists when infinite graphs are allowed, as follows.…”

“…The isomorphism problem of finitely generated Coxeter groups in these classes is well studied; see [Rad03], [MW02], [Bah05], and [Muh05], respectively. In particular, these results give certain conditions for the Coxeter graph of a Coxeter group in each class to be unique (up to isomorphism).…”