On automorphisms and splittings of special groups

TL;DR

This paper explores the automorphism groups of special groups, establishing conditions under which these groups are infinite and analyzing their splittings and Dehn twists using actions on -trees.

Contribution

It introduces criteria for the infiniteness of outer automorphism groups of special groups and examines how splittings relate to Dehn twists and coarse median structures.

Findings

Outer automorphism group is infinite iff certain splittings and Dehn twists exist.

Coarse-median preserving automorphisms are characterized by specific splittings.

Analysis of actions on -trees reveals the structure of stabilizers and automorphisms.

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.