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Acylindrical and strong accessibility 10.17863/CAM.81049 Hill, Michael Wilton, Henry 0000000163699478 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. University of Cambridge 2021-06-26 Thesis Doctoral Doctor of Philosophy (PhD) eng https://www.repository.cam.ac.uk/handle/1810/333633 https://www.repository.cam.ac.uk/bitstreams/13fc551e-8c09-4dc3-9889-72ee25dce15a/download e09c5a795b16d911a2f98d538f39122d http://purl.org/NET/rdflicense/allrightsreserved Group Theory Geometric Group Theory Bass-Serre Theory Accessibility of Groups