On automorphisms and splittings of special groups

We initiate the study of outer automorphism groups of special groups G, in the Haglund–Wise sense. We show that 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 Out ₂₌₏ (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 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.