Generalizations of quasiconvexity for finitely generated groups

Kim, Heejoung

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https://hdl.handle.net/2142/110434 Copy

Description

Title

Generalizations of quasiconvexity for finitely generated groups

Author(s)

Kim, Heejoung

Issue Date

2021-04-15

Director of Research (if dissertation) or Advisor (if thesis)

Kapovich, Ilya

Doctoral Committee Chair(s)

Leininger, Christopher J

Committee Member(s)

• Dunfield, Nathan
• Schupp, Paul

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

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

Abstract

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.

Graduation Semester

2021-05

Type of Resource

Thesis

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

Copyright and License Information

Copyright 2021 Heejoung Kim

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