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 Γ \Gamma of a right-angled Coxeter group W Γ 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 F_2 as a quotient. When Out ⁡ ( W Γ ) \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 Γ \Gamma contains no SIL.