On automorphisms and splittings of special groups

Abstract: We initiate the study of outer automorphism groups of special groups $G$, in the Haglund-Wise sense. We show that ${\rm Out}(G)$ is infinite if and only if $G$ splits over a co-abelian subgroup of a centraliser and there exists an infinite-order "generalised Dehn twist". Similarly, the coarse-median preserving subgroup ${\rm Out}_{\rm cmp}(G)$ is infinite if and only if $G$ splits over an actual centraliser and there exists an infinite-order coarse-median-preserving generalised Dehn twist. The proof is based on constructing and analysing non-small, stable $G$-actions on $\mathbb{R}$-trees whose arc-stabilisers are centralisers or closely related subgroups. Interestingly, tripod-stabilisers can be arbitrary centralisers, and thus are large subgroups of $G$. As a result of independent interest, we determine when generalised Dehn twists associated to splittings of $G$ preserve the coarse median structure.