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

Abstract:

We generalize 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 $\operatorname {Out}(W_\Gamma )$ 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.