{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/333633"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/333633","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"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.","abstract_html":"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.","abstract_has_math":false,"creators":["Hill, Michael"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Wilton, Henry"],"committee_chairs":[],"committee_members":[],"year":2021,"date_issued":"2021-06-26","date_published":"2021-06-26","updated_at":"2026-07-24T01:33:25Z","subjects":["Group Theory","Geometric Group Theory","Bass-Serre Theory","Accessibility of Groups"],"languages":["eng"],"rights":[],"rights_urls":["http://purl.org/NET/rdflicense/allrightsreserved"],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.81049","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Wilton, Henry"]},{"key":"dc:creator","label":"Author","values":["Hill, Michael"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2021-06-26"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/333633"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Group Theory","Geometric Group Theory","Bass-Serre Theory","Accessibility of Groups"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["http://purl.org/NET/rdflicense/allrightsreserved"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.17863/CAM.81049"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://www.repository.cam.ac.uk/bitstreams/13fc551e-8c09-4dc3-9889-72ee25dce15a/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["e09c5a795b16d911a2f98d538f39122d"]},{"key":"dc:title","label":"Title","values":["Acylindrical and strong accessibility"]}]}],"canonical_facts":{"dc:contributor.advisor":["Wilton, Henry"],"dc:creator":["Hill, Michael"],"dc:date.issued":["2021-06-26"],"dc:description.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."],"dc:format.checksum.md5":["e09c5a795b16d911a2f98d538f39122d"],"dc:identifier.doi":["10.17863/CAM.81049"],"dc:identifier.uri":["https://www.repository.cam.ac.uk/bitstreams/13fc551e-8c09-4dc3-9889-72ee25dce15a/download"],"dc:language":["eng"],"dc:publisher.institution":["University of Cambridge"],"dc:relation.isreferencedby.uri":["https://www.repository.cam.ac.uk/handle/1810/333633"],"dc:rights":["http://purl.org/NET/rdflicense/allrightsreserved"],"dc:subject":["Group Theory","Geometric Group Theory","Bass-Serre Theory","Accessibility of Groups"],"dc:title":["Acylindrical and strong accessibility"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T01:33:25Z"}