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

TL;DR

This work develops a robust framework connecting relative hyperbolicity, quasiconvexity, and profinite separability to study virtual joins and products. By introducing metric criteria and leveraging the profinite topology, the authors prove the existence of quasiconvex virtual joins for pairs of finitely generated relatively quasiconvex subgroups in QCERF relatively hyperbolic groups, without requiring parabolic compatibility, and establish separability of products under generalized RZ_s assumptions. These results yield broad separability conclusions for products in important classes such as limit groups, Kleinian groups, and balanced graphs of free groups with cyclic edge groups, and provide a pathway to geometric finiteness in virtual joins. The methods combine shortcutting of broken-line paths, control of backtracking, and novel double-coset separability criteria in amalgams, yielding new insights into profinite closures of quasiconvex subgroups and the structure of products in relatively hyperbolic groups.

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 (e.g., 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.

Figures (8)

• Figure 1: We obtain a different path representative for $g$ by replacing $p_i$ and $p_{i+1}$ with geodesics from $f_{i-1}$ to $z$ to $f_{i+1}$.
• Figure 2: Illustration of Lemma \ref{['lem:one_comp_in_cusp_is_bounded']}.
• Figure 3: Illustration of Lemma \ref{['lem:end_sides_constr']}.
• Figure 4: Illustration of Lemma \ref{['lem:(c3)->vertex_constr']}.
• Figure 5: The new path $p'$ constructed in Proposition \ref{['prop:multitracking_path']}. The dotted lines between $p$ and $p'$ are paths whose labels represent elements of $S$.

Theorems & Definitions (209)

• Definition 1.1: QCERF
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Definition 1.5: Almost compatible parabolics
• Corollary 1.6
• Definition 1.7: RZ$_s$ and product separability
• Theorem 1.8
• Corollary 2.1
• proof