A characterization of virtually cyclic outer automorphism groups of right-angled Coxeter groups

TL;DR

This paper characterizes when the outer automorphism group of right-angled Coxeter groups is virtually cyclic, based on combinatorial properties of the defining graph related to separating intersections of links.

Contribution

It establishes conditions involving SILs, STILs, and FSILs in the graph that determine when the outer automorphism group is virtually cyclic.

Findings

Outer automorphism group is virtually Z under certain SIL conditions.

Characterization applies to both connected and disconnected graphs.

Provides a combinatorial criterion for virtual cyclicity in right-angled Coxeter groups.

Abstract

Existing research gives conditions for when the outer automorphism group of a graph product of primary cyclic groups $W_\Gamma$ is finite, virtually abelian, or large. We seek to prove a set of conditions for when this outer automorphism group is virtually cyclic. To this end, we study the finite index subgroup $\text{Out}^0(W_\Gamma)$, which is generated by specific partial conjugations. The presence or absence of Coxeter and non-Coxeter separating intersections of links (SILs), separating triple intersections of links (STILs), and flexible separating intersections of links (FSILs) in $\Gamma$ determines algebraic properties of $\text{Out}^0(W_\Gamma)$. We identify each SIL with a pair of partial conjugations in $\text{Out}^0(W_\Gamma)$ and place restrictions on the SILs in $\Gamma$ to ensure that $\text{Out}^0(W_\Gamma)$ is virtually $\mathbb{Z}$ both when $\Gamma$ is connected or disconnected. In particular, this applies to the study of right-angled Coxeter groups. This paper is a slightly shorter version of the author's master's thesis from Tufts University.