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arXiv · 2310.00512

Structure of quasiconvex virtual joins

Lawk Mineh
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Abstract

Let $G$ be a relatively hyperbolic group and let $Q$ and $R$ be relatively quasiconvex subgroups. It is known that there are many pairs of finite index subgroups $Q' \leqslant_f Q$ and $R' \leqslant_f R$ such that the subgroup join $\langle Q', R' \rangle$ is also relatively quasiconvex, given suitable assumptions on the profinite topology of $G$. We show that the intersections of such joins with maximal parabolic subgroups of $G$ are themselves joins of intersections of the factor subgroups $Q'$ and $R'$ with maximal parabolic subgroups of $G$. As a consequence, we show that quasiconvex subgroups whose parabolic subgroups are almost compatible have finite index subgroups whose parabolic subgroups are compatible, and provide a combination theorem for such subgroups.

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Lawk Mineh. 2025-04-02. Structure of quasiconvex virtual joins. https://arxiv.org/abs/2310.00512

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