On automorphisms and splittings of special groups. Issue 2 (10th February 2023)

• Record Type: Journal Article
• Authors: Fioravanti, Elia
• Is Part Of: Compositio mathematica. Volume 159:Issue 2(2023)
• Page Start: 232
• Page End: 305
• Publication Date: 2023-02-10
• Subjects: special group -- Dehn twist -- coarse-median preserving -- outer automorphism; 20F65 -- 20F67 -- 20F28 -- 20E06 -- 20E08 -- 57M07; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X22007850
• Languages: English
• ISSNs: 0010-437X

Abstract

Abstract : We initiate the study of outer automorphism groups of special groups $G$, in the Haglund–Wise sense. We show that $\operatorname {Out}(G)$ 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 $\operatorname {Out}_{\rm cmp}(G)$ 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.