Quasiconvexity of virtual joins and separability of products in relatively hyperbolic groups
Quasiconvexity of virtual joins and separability of products in relatively hyperbolic groups
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 $Q,R \leqslant G$ we prove the existence of finite index subgroups $Q'\leqslant_f Q$ and $R' \leqslant_f R$ such that the join $\langle Q',R'\rangle$ 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.
Relatively hyperbolic groups, relatively quasiconvex subgroups, virtual joins, double coset separability, product separability, limit groups, Kleinian groups
Minasyan, Ashot
3de640f5-d07b-461f-b130-5b1270bfdb3d
Mineh, Lawk
01bd774a-7615-4223-89c5-aa798e421853
Minasyan, Ashot
3de640f5-d07b-461f-b130-5b1270bfdb3d
Mineh, Lawk
01bd774a-7615-4223-89c5-aa798e421853
Minasyan, Ashot and Mineh, Lawk
(2023)
Quasiconvexity of virtual joins and separability of products in relatively hyperbolic groups.
Algebraic & Geometric Topology.
(In Press)
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 $Q,R \leqslant G$ we prove the existence of finite index subgroups $Q'\leqslant_f Q$ and $R' \leqslant_f R$ such that the join $\langle Q',R'\rangle$ 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.
Text
Separability_of_quasiconvex_products_in_relatively_hyperbolic_groups-accepted
- Accepted Manuscript
More information
Accepted/In Press date: 1 October 2023
Keywords:
Relatively hyperbolic groups, relatively quasiconvex subgroups, virtual joins, double coset separability, product separability, limit groups, Kleinian groups
Identifiers
Local EPrints ID: 483012
URI: http://eprints.soton.ac.uk/id/eprint/483012
ISSN: 1472-2747
PURE UUID: c89ec058-9625-47db-ae89-feef046b35e2
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Date deposited: 19 Oct 2023 16:45
Last modified: 18 Mar 2024 03:08
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