On compactness properties of subgroups

Arora, Shivam (2022) On compactness properties of subgroups. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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Abstract

The class of locally compact groups has been widely studied in group theory, representation theory, and harmonic analysis. There is a current program of extending geometric techniques used in the study of discrete groups to this larger class [Wil94, KM08,CCMT15,CdlH16]. This thesis is part of that program. We use geometric methods to study the compactness properties of subgroups in the class of topological groups containing a compact open subgroup. This class includes discrete groups, profinite groups, and totally disconnected locally compact groups as subclasses. In the first project, we study discrete hyperbolic groups. Finitely presented subgroups of hyperbolic groups are not necessarily hyperbolic; the first examples of this phenomenon were constructed by Brady [Bra99]. In contrast, for hyperbolic groups of integral cohomological dimension at most two, finitely presented subgroups are hyperbolic; this is a result of Gersten [Ger96b]. We extend this result to hyperbolic groups with rational cohomological dimension bounded by two. This applies to examples of groups constructed by Bestvina and Mess, fully describing the nature of their finitely presented subgroups, which was previously unknown. In the second project, we extend Gersten’s result for totally disconnected locally compact (TDLC) groups. In particular, we prove that closed compactly presented subgroups of hyperbolic TDLC groups of discrete rational cohomological dimension bounded by two are hyperbolic. We also characterize hyperbolic TDLC groups in terms of isoperimetric inequalities and study small cancellation quotients of amalgamated free products of profinite groups over open subgroups. In the last project, we study coherence of topological groups. A group is coherent if every compactly generated subgroup is compactly presented. We prove that amalgamated free products of coherent groups over compact open subgroups are coherent. We also show that certain small cancellation quotients of these groups are also coherent, generalizing a result of McCammond and Wise [MW05]. In order to prove the main results, we study relative hyperbolicity for topological groups containing compact open subgroups with respect to finite collection of open subgroups, and extend some results of Osin [Osi06].

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