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

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.