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Structure of quasiconvex virtual joins

Lawk Mineh

TL;DR

The paper addresses how joins of relatively quasiconvex subgroups sit inside QCERF relatively hyperbolic groups, focusing on the parabolic structure of those joins. It develops a technical framework—shortcutting of broken lines, tamable path representatives, and careful control of parabolic intersections via profinite/separability methods—to show that the intersections with maximal parabolic subgroups are, up to conjugacy, joins of the corresponding intersections. It proves that almost compatible parabolics can be virtually promoted to compatible parabolics and establishes a robust combination-type theorem for such subgroups, including an amalgamated-to-free product transition under suitable hypotheses. The results provide a structural description of virtual joins, enable amalgamation and virtual compatibility outcomes, and extend to strongly/full relatively quasiconvex subgroups, with implications for the hyperbolicity of the resulting joins and their parabolic configurations.

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.

Structure of quasiconvex virtual joins

TL;DR

The paper addresses how joins of relatively quasiconvex subgroups sit inside QCERF relatively hyperbolic groups, focusing on the parabolic structure of those joins. It develops a technical framework—shortcutting of broken lines, tamable path representatives, and careful control of parabolic intersections via profinite/separability methods—to show that the intersections with maximal parabolic subgroups are, up to conjugacy, joins of the corresponding intersections. It proves that almost compatible parabolics can be virtually promoted to compatible parabolics and establishes a robust combination-type theorem for such subgroups, including an amalgamated-to-free product transition under suitable hypotheses. The results provide a structural description of virtual joins, enable amalgamation and virtual compatibility outcomes, and extend to strongly/full relatively quasiconvex subgroups, with implications for the hyperbolicity of the resulting joins and their parabolic configurations.

Abstract

Let be a relatively hyperbolic group and let and be relatively quasiconvex subgroups. It is known that there are many pairs of finite index subgroups and such that the subgroup join is also relatively quasiconvex, given suitable assumptions on the profinite topology of . We show that the intersections of such joins with maximal parabolic subgroups of are themselves joins of intersections of the factor subgroups and with maximal parabolic subgroups of . 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.
Paper Structure (13 sections, 45 theorems, 74 equations, 4 figures)

This paper contains 13 sections, 45 theorems, 74 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Notation and terminology
  5. Relatively hyperbolic groups
  6. Profinite topology
  7. Controlling parabolic subgroups in relatively hyperbolic groups
  8. Quasigeodesics and shortcuttings
  9. Shortcuttings for parabolic paths
  10. Path representatives
  11. Metric conditions and path representatives of minimal type
  12. Reduction to the short conjugator case
  13. Proofs of main results

Key Result

Theorem 1.1

Let $G$ be a finitely generated group. Suppose that $G$ is QCERF hyperbolic relative to a collection of double coset separable subgroups and let $Q, R \leqslant G$ be finitely generated relatively quasiconvex subgroups. Then there are finite index subgroups $Q' \leqslant_f Q$ and $R' \leqslant_f R$

Figures (4)

  • Figure 1: Illustration of Lemma \ref{['lem:h_conn_to_ek']}.
  • Figure 2: Illustration of proof of Lemma \ref{['lem:h_conn_to_ek']}.
  • Figure 3: Illustration of Lemma \ref{['lem:parab_short_one_ek']}.
  • Figure 4: Illustration of Lemma \ref{['lem:pass_to_q_conj']}.

Theorems & Definitions (92)

  • Theorem 1.1: MinMin
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Remark 1.7
  • Definition 2.1: Broken line
  • Remark 2.2
  • Definition 2.3
  • ...and 82 more