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

TL;DR

This paper extends fundamental concepts of hyperbolic groups to totally disconnected locally compact (TDLC) groups, proving combination theorems, constructing Gromov boundaries, and establishing Cannon-Thurston maps in this broader setting.

Contribution

It introduces a combination theorem for acylindrical graphs of hyperbolic TDLC groups and constructs Cannon-Thurston maps for these groups, generalizing prior results.

Findings

Proved a combination theorem for acylindrical graphs of hyperbolic TDLC groups.

Constructed explicit Gromov boundaries for fundamental groups of graphs of groups.

Established the existence of Cannon-Thurston maps for hyperbolic TDLC groups.

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.