On automorphisms and splittings of special groups
Elia Fioravanti

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 , in the Haglund-Wise sense. We show that is infinite if and only if splits over a co-abelian subgroup of a centraliser and there exists an infinite-order "generalised Dehn twist". Similarly, the coarse-median preserving subgroup is infinite if and only if 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 -actions on -trees whose arc-stabilisers are centralisers or closely related subgroups. Interestingly, tripod-stabilisers can be arbitrary centralisers, and thus are large subgroups of . As a result of independent interest, we determine when generalised Dehn twists associated to splittings of preserve the…
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Taxonomy
TopicsGeometric and Algebraic Topology · Advanced Operator Algebra Research · Finite Group Theory Research
