Quasi-Isometries for certain Right-Angled Coxeter Groups

TL;DR

This paper constructs a JSJ tree for certain right-angled Coxeter groups using their defining graphs, providing a quasi-isometry invariant that helps classify these groups up to quasi-isometry.

Contribution

It generalizes the construction of the JSJ tree for RACGs and refines invariants to classify groups based on their defining graphs and decompositions.

Findings

The JSJ tree $T_c$ has two-ended edge stabilizers iff the defining graph lacks a subdivided $K_4$.

The structure invariant is a quasi-isometry invariant determined by the defining graph.

Refinement of the invariant yields a complete quasi-isometry classification when no rigid vertices are present.

Abstract

We construct the JSJ tree of cylinders $T_c$ for finitely presented, one-ended, two-dimensional right-angled Coxeter groups (RACGs) splitting over two-ended subgroups in terms of the defining graph of the group, generalizing the visual construction by Dani and Thomas given for hyperbolic RACGs. Additionally, we prove that $T_c$ has two-ended edge stabilizers if and only if the defining graph does not contain a certain subdivided $K_4$. By use of the structure invariant of $T_c$ introduced by Cashen and Martin, we obtain a quasi-isometry-invariant of these RACGs, essentially determined by the defining graph. Furthermore, we refine the structure invariant to make it a complete quasi-isometry-invariant in case the JSJ decomposition of the RACG does not have any rigid vertices.