Strong accessibility of Coxeter groups over minimal splittings

TL;DR

This paper proves that Coxeter groups are strongly accessible over their minimal splittings, and that their decompositions over such splittings preserve the Coxeter structure in vertex and edge groups.

Contribution

It establishes the strong accessibility of Coxeter groups over minimal splittings and shows that their decompositions maintain Coxeter properties.

Findings

Coxeter groups are strongly accessible over minimal splittings.

Vertex and edge groups in decompositions are Coxeter groups.

Minimal splitting subgroups of Coxeter groups are finitely generated.

Abstract

Given a class of groups C, a group G is strongly accessible over C if there is a bound on the number of terms in a sequence L(1), L(2), ..., L(n) of graph of groups decompositions of G with edge groups in C such that L(1) is the trivial decomposition (with 1-vertex) and for i>1, L(i) is obtained from L(i-1) by non-trivially and compatibly splitting a vertex group of L(i-1) over a group in C, replacing this vertex group by the splitting and then reducing. If H and K are subgroups of a group G then H is smaller than K if H intersect K has finite index in H and infinite index in K. The minimal splitting subgroups of G, are the subgroups H of G, such that G splits non-trivially (as an amalgamated product or HNN-extension) over H and for any other splitting subgroup K of W, K is not smaller than H. When G is a finitely generated Coxeter group, minimal splitting subgroups are always finitely generated. Minimal splittings are explicitly or implicitly important aspects of Dunwoody's work on accessibility and the JSJ results of Rips-Sela, Dunwoody-Sageev and Mihalik. Our main results are that Coxeter groups are strongly accessible over minimal splittings and if L is an irreducible graph of groups decomposition of a Coxeter group with minimal splitting edge groups, then the vertex and edge groups of L are Coxeter.