JSJ decompositions of Coxeter groups over FA subgroups

TL;DR

This paper proves that all finite rank Coxeter groups have a visual JSJ decomposition over FA subgroups, simplifying the twist conjecture by focusing on indecomposable Coxeter systems.

Contribution

It introduces a visual JSJ decomposition framework for Coxeter groups over FA subgroups and reduces the twist conjecture to indecomposable cases.

Findings

Every finite rank Coxeter system admits a visual JSJ decomposition over FA subgroups.

The twist conjecture can be reduced to Coxeter systems indecomposable under certain amalgamations.

Provides a new structural understanding of Coxeter groups in relation to FA subgroups.

Abstract

A group G has property FA if G fixes a point of every tree on which G acts without inversions. We prove that every Coxeter system of finite rank has a visual JSJ decomposition over subgroups with property FA. As an application, we reduce the twist conjecture to Coxeter systems that are indecomposable with respect to amalgamated products over visual subgroups with property FA.