JSJ Decompositions of Coxeter Groups

TL;DR

This paper develops a unique, algorithmic JSJ-decomposition for Coxeter groups over virtually abelian subgroups, revealing the group's structure through visual, vertex, and edge groups.

Contribution

It introduces a novel, visual JSJ-decomposition for Coxeter groups, with a constructive, algorithmic process and a detailed classification of vertex groups.

Findings

The decomposition is unique for Coxeter groups.

Vertex groups are either virtually surface, free, or abelian.

The construction is algorithmic and visual.

Abstract

The idea of "JSJ-decompositions" for 3-manifolds began with work of Waldhausen and was developed later through work of Jaco, Shalen and Johansen. It was shown that there is a finite collection of 2-sided, incompressible tori that separate a given closed irreducible 3-manifold into pieces with strong topological structure. Sela introduced the idea of JSJ-decompositions for groups, an idea that has flourished in a variety of directions. The general idea is to consider a certain class X of groups and splittings of groups in X by groups in another class Y. E.g. Rips and Sela considered splittings of finitely presented groups by infinite cyclic groups. For an arbitrary group G in X the goal is to produce a unique graph of groups decomposition T of G with edge groups in Y so that T reveals all graph of groups decompositions of G with edge groups in Y. More specifically, if V is a vertex group of T then either there is no Y-group that splits both G and V, or V has a special "surface group-like" structure. It is standard to call vertex groups of the second type "orbifold groups". For a finitely generated Coxeter system (W,S) we produce a reduced JSJ-decomposition T for splittings of W over virtually abelian subgroups. We show T is unique with each vertex and edge group generated by a subset of S (and so T is "visual"). The construction of T is algorithmic. If V, a subset of S, generates an orbifold vertex group of T then V is the disjoint union of K and M, where < M > is virtually abelian, < K > is virtually a closed surface group or virtually free and < V > is the direct product of < M > and < K >.