A characterization of virtually cyclic outer automorphism groups of 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.