Acylindrical and strong accessibility

Abstract

Weidmann has produced a bound on the number of edges of a graph of groups splitting for when a finitely generated group acts on a tree (𝑘,𝐶)-acylindrically [25]. In the same paper Weidmann conjectures a common generalisation between their result and a theorem of Bestvina and Feighn [2]; which provides a similar bound for finitely generated groups acting on a tree with small edge stabilisers. We will produce an example which shows this conjecture is false. We then extend Weidmann’s result to actions which are 𝑘-acylindrical except on some set of subgroups with finite height. We then apply this result to a couple of specific cases. The first gives us a bound for actions of hyperbolic groups which are 𝑘-acylindrical on non virtually-cyclic subgroups. The second give a bound for a RAAG acting 𝑘-acylindrically on non-abelian subgroups. We also provide a sharp bound for finitely generated groups acting 𝑘-acylindrically.

We also touch on the subject of strong accessibility. In particular we give an account of a theorem by Louder and Touikan [19] which shows that many hierarchies consisting of slender JSJ-decompositions are finite; in particular JSJ-hierarchies of 2-torsion-free hyperbolic groups are always finite.