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 $\mathrm{Out}(G)$ 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 ${\mathrm{Out}}_{\mathrm{cmp}}(G)$ 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.

67 pages, 1 figure; added references to introduction, corrected some typos. To appear in Compositio Mathematica

References in corpus (3)

• Poc Sets, Median Algebras and Group Actions
• Coarse-median preserving automorphisms
• Convex cores for actions on finite-rank median algebras

Cited by in corpus (3)

• Coarse-median preserving automorphisms
• Coarse cubical rigidity
• Divisible cube complexes and finite-order automorphisms of RAAGs