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Combination of locally quasiconvex hyperbolic TDLC groups and Cannon-Thurston maps

Swarnali Datta, Arunava Mandal, Ravi Tomar

TL;DR

The paper generalizes key geometric-group-theoretic constructions to hyperbolic totally disconnected locally compact (TDLC) groups. By employing Dahmani’s boundary framework and Carette–Dreesen’s convergence characterization, it establishes a robust combination theorem for finite acylindrical graphs of hyperbolic TDLC groups and provides an explicit description of the Gromov boundary of the resulting fundamental group. It then extends the theory of locally quasiconvex subgroups, height, and Cannon–Thurston maps to this TDLC setting, proving a combination theorem for locally quasiconvex vertex groups and the existence of CT maps for extensions of hyperbolic TDLC groups. A parallel development shows the existence of quasiisometric sections for short exact sequences, enabling ladder constructions that underpin the CT-map results. Collectively, the work broadens the discrete hyperbolic theory to TDLC groups, with structural consequences for boundary dynamics, subgroup geometry, and extensions, and it unifies several strands of convergence group theory in a non-discrete context.

Abstract

In this article, we study acylindrical graphs of groups, local quasiconvexity, and Cannon-Thurston maps in the setting of totally disconnected locally compact (TDLC) hyperbolic groups, extending several fundamental notions and results from discrete hyperbolic groups to this broader context. Leveraging Dahmani's technique and a topological characterization of hyperbolic TDLC groups in terms of uniform convergence groups given by Carette-Dreesen, we prove a combination theorem for an acylindrical graph of hyperbolic TDLC groups and give an explicit construction of the Gromov boundary of the fundamental group of the given graph of groups. Using the description of the Gromov boundary, we prove our main result: a combination theorem for an acylindrical graph of locally quasiconvex hyperbolic TDLC groups. Further, we generalise the work of Mosher, proving the existence of quasiisometric sections for a given short exact sequence of hyperbolic TDLC groups. This leads us to prove the existence of a Cannon-Thurston map for a normal hyperbolic subgroup of a hyperbolic TDLC group, generalising a theorem of Mj.

Combination of locally quasiconvex hyperbolic TDLC groups and Cannon-Thurston maps

TL;DR

The paper generalizes key geometric-group-theoretic constructions to hyperbolic totally disconnected locally compact (TDLC) groups. By employing Dahmani’s boundary framework and Carette–Dreesen’s convergence characterization, it establishes a robust combination theorem for finite acylindrical graphs of hyperbolic TDLC groups and provides an explicit description of the Gromov boundary of the resulting fundamental group. It then extends the theory of locally quasiconvex subgroups, height, and Cannon–Thurston maps to this TDLC setting, proving a combination theorem for locally quasiconvex vertex groups and the existence of CT maps for extensions of hyperbolic TDLC groups. A parallel development shows the existence of quasiisometric sections for short exact sequences, enabling ladder constructions that underpin the CT-map results. Collectively, the work broadens the discrete hyperbolic theory to TDLC groups, with structural consequences for boundary dynamics, subgroup geometry, and extensions, and it unifies several strands of convergence group theory in a non-discrete context.

Abstract

In this article, we study acylindrical graphs of groups, local quasiconvexity, and Cannon-Thurston maps in the setting of totally disconnected locally compact (TDLC) hyperbolic groups, extending several fundamental notions and results from discrete hyperbolic groups to this broader context. Leveraging Dahmani's technique and a topological characterization of hyperbolic TDLC groups in terms of uniform convergence groups given by Carette-Dreesen, we prove a combination theorem for an acylindrical graph of hyperbolic TDLC groups and give an explicit construction of the Gromov boundary of the fundamental group of the given graph of groups. Using the description of the Gromov boundary, we prove our main result: a combination theorem for an acylindrical graph of locally quasiconvex hyperbolic TDLC groups. Further, we generalise the work of Mosher, proving the existence of quasiisometric sections for a given short exact sequence of hyperbolic TDLC groups. This leads us to prove the existence of a Cannon-Thurston map for a normal hyperbolic subgroup of a hyperbolic TDLC group, generalising a theorem of Mj.
Paper Structure (24 sections, 50 theorems, 23 equations)

This paper contains 24 sections, 50 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Acylindrical graphs of hyperbolic TDLC groups
  3. Combination of locally quasiconvex hyperbolic TDLC groups
  4. Cannon-Thurston maps for extensions of hyperbolic TDLC groups
  5. Background
  6. Coarse geometry
  7. Graphs of topological groups
  8. Basic properties of Cayley-Abels graphs
  9. Hyperbolic TDLC groups
  10. Convergence TDLC groups
  11. Cannon-Thurston maps
  12. Quasiconvex subgroups of hyperbolic TDLC groups and their limit sets
  13. Application to CT maps
  14. Limit set of quasiconvex subgroups
  15. Height of a subgroup of a TDLC group
  16. ...and 9 more sections

Key Result

Theorem 1.2

Let $(\mathcal{G},\mathcal{Z})$ be a finite graph of groups with fundamental group $G$ such that the following hold: Then, $G$ is a hyperbolic TDLC group, and the vertex groups are quasiconvex in $G$.

Theorems & Definitions (103)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Remark 2.5
  • ...and 93 more