Quasiconvexity of virtual joins and separability of products in relatively hyperbolic groups

Minasyan, Ashot; Mineh, Lawk

doi:10.2140/agt.2025.25.399

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Quasiconvexity of virtual joins and separability of products in relatively hyperbolic groups

Ashot Minasyan and Lawk Mineh

Algebraic & Geometric Topology 25 (2025) 399–488

DOI: 10.2140/agt.2025.25.399

Abstract

A relatively hyperbolic group $G$ is said to be QCERF if all finitely generated relatively quasiconvex subgroups are closed in the profinite topology on $G$ .

Assume that $G$ is a QCERF relatively hyperbolic group with double coset separable (eg virtually polycyclic) peripheral subgroups. Given any two finitely generated relatively quasiconvex subgroups $Q, R \leq G$ we prove the existence of finite-index subgroups $Q^{'} \leq_{f} Q$ and $R^{'} \leq_{f} R$ such that the join $⟨ Q^{'}, R^{'} ⟩$ is again relatively quasiconvex in $G$ . We then show that, under the minimal necessary hypotheses on the peripheral subgroups, products of finitely generated relatively quasiconvex subgroups are closed in the profinite topology on $G$ . From this we obtain the separability of products of finitely generated subgroups for several classes of groups, including limit groups, Kleinian groups and balanced fundamental groups of finite graphs of free groups with cyclic edge groups.

Keywords

relatively hyperbolic groups, relatively quasiconvex subgroups, virtual joins, double coset separability, product separability, limit groups, Kleinian groups

Mathematical Subject Classification

Primary: 20E26, 20F65, 20F67

Secondary: 20H10

References

Publication

Received: 28 September 2022

Revised: 21 April 2023

Accepted: 1 October 2023

Published: 24 March 2025

Authors

Ashot Minasyan
School of Mathematical Sciences University of Southampton Southampton United Kingdom

Lawk Mineh
Mathematical Institute University of Bonn Bonn Germany

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Volume 25, issue 1 (2025)