On conjugacy separability of some Coxeter groups and parabolic-preserving automorphisms

TL;DR

This paper proves conjugacy separability for certain even Coxeter groups, explores automorphisms preserving parabolic subgroups, and derives implications for their automorphism groups.

Contribution

It establishes conjugacy separability for even Coxeter groups with specific diagram restrictions and analyzes automorphisms preserving parabolic subgroups.

Findings

Even Coxeter groups without (4,4,2) triangles are conjugacy separable.

Automorphisms preserving conjugacy classes of short elements are inner.

Results impact understanding of outer automorphism groups of Coxeter groups.

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.