# Outer automorphism groups of right-angled Coxeter groups are either large or virtually abelian

## TL;DR

This paper characterizes when the outer automorphism groups of right-angled Coxeter groups are large or virtually abelian based on graph properties, extending to graph products of finite abelian groups.

## Contribution

It introduces generalized criteria for the largeness of outer automorphism groups and establishes a dichotomy for these groups based on graph conditions.

## Key findings

- Outer automorphism groups are large if and only if the defining graph contains a SIL.
- If not large, the outer automorphism group is virtually abelian.
- Groups are property (T) only if they are finite, i.e., the graph contains no SIL.

## Abstract

We generalise the notion of a separating intersection of links (SIL) to give necessary and sufficient criteria on the defining graph $\Gamma$ of a right-angled Coxeter group $W_\Gamma$ so that its outer automorphism group is large: that is, it contains a finite index subgroup that admits the free group $F_2$ as a quotient. When $Out(W_\Gamma)$ is not large, we show it is virtually abelian. We also show that the same dichotomy holds for the outer automorphism groups of graph products of finite abelian groups. As a consequence, these groups have property (T) if and only if they are finite, or equivalently $\Gamma$ contains no SIL.