Quasiconvexity and separability in relatively hyperbolic groups

Abstract

In this thesis, we aim to understand the behaviour of quasiconvexity under the basic operation of taking joins of subgroups. We will also study the relation between quasiconvexity and residual properties of relatively hyperbolic groups.

In particular, suppose that G is a relatively hyperbolic group, and let Q and R be relatively quasiconvex subgroups of G. We provide sufficient conditions for the join 〈Q', R'〉 of subgroups Q' of Q and R' of R to be relatively quasiconvex. Further, we determine the structure of the maximal parabolic subgroups of 〈Q', R'〉 in this setting.

We show that, given suitable assumptions on the profinite topology of G, these conditions can be arranged to hold for sufficiently deep finite index subgroups Q' of Q and R' of R. As a consequence, we show that 〈Q', R'〉 decomposes as an amalgamated free product when the parabolic subgroups of Q and R are almost compatible.

Finally, we show that if G is hyperbolic relative to product separable subgroups, then the product of any finitely generated quasiconvex subgroups is separable in G. We record applications of this to various classes of nonpositively curved groups.

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Identifiers

ORCID for Ashot Minasyan: orcid.org/0000-0002-4986-2352

ORCID for Ian Leary: orcid.org/0000-0001-8300-4979

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