Acylindrical and strong accessibility

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.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2021

Author dc:creator

• Hill, Michael

Advisor dc:contributor.advisor

• Wilton, Henry

• Group Theory
• Geometric Group Theory
• Bass-Serre Theory
• Accessibility of Groups

Licence

• http://purl.org/NET/rdflicense/allrightsreserved

Language dc:language

eng

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.81049

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/333633