Generalizations of quasiconvexity for finitely generated groups

For a word-hyperbolic group G, the notion of quasiconvexity of a finitely generated subgroup H of G is independent of the choices of finite generating sets for G and H, and is equivalent to H being quasi- isometrically embedded in G. However, beyond word-hyperbolic groups, the notion of quasiconvexity is not as useful. For a finitely generated group, there are two recent generalizations of the notion of a quasiconvex subgroup of a word-hyperbolic group, a “stable” subgroup and a “Morse” subgroup. Durham and Taylor [33] defined stability and proved stability is equivalent to convex cocompactness in mapping class groups. Another natural generalization of quasiconvexity is given by the notion of a Morse or strongly quasiconvex subgroup of a finitely generated group, studied by Tran [92] and Genevois [37]. For an arbitrary finitely generated group, an infinite subgroup is stable if and only if the subgroup is Morse and hyperbolic. We prove that two properties of being Morse of infinite index and stable coincide for a subgroup of infinite index in the mapping class group of an oriented, connected, finite type surface with negative Euler characteristic [67]. Finding algorithms for the detection and decidability of various properties of groups is a fundamental theme in geometric group theory. For a word-hyperbolic group G, Kapovich [55] provided a partial algorithm which, on input a finite set S of G, halts if S generates a quasiconvex subgroup of G and runs forever otherwise. In this thesis, we give various detection and decidability algorithms for stability and Morseness of mapping class groups, right-angled Artin groups, toral relatively hyperbolic groups which contains finitely generated groups discriminated by a locally quasiconvex torsion-free hyperbolic group (for example, ordinary limit groups) [68]. Also, we provide a partial algorithm which, for a finite subset S of a toral relatively hyperbolic group, terminates if S generates a relatively quasiconvex subgroup of G, equivalently, the subgroup generated by S is undistorted in G.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2021

Author dc:creator

• Kim, Heejoung

Contributors dc:contributor

• Kapovich, Ilya
• Leininger, Christopher J
• Dunfield, Nathan
• Schupp, Paul

• Hyperbolic group
• Quasiconvex subgroup
• Stable subgroup
• Morse subgroup
• Mapping class group
• Right-angled Artin group
• Toral relatively hyperbolic group

Statement dc:rights

• Copyright 2021 Heejoung Kim

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/110434

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/110434