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

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.

💡 Research Summary

This paper develops a comprehensive combination theory for hyperbolic totally disconnected locally compact (TDLC) groups, extending three central notions—acylindrical graphs of groups, local quasiconvexity, and Cannon‑Thurston (CT) maps—from the classical discrete setting to the broader TDLC context.

The authors first adapt the concept of an acylindrical action to TDLC groups. For a graph of topological groups ((G,Z)) with Bass‑Serre tree (T), the action of the fundamental group on (T) is called (k)-acylindrical if the pointwise stabilizer of any geodesic of length greater than (k) is compact. This mirrors Sela’s definition for discrete groups but incorporates the compactness requirements inherent to TDLC groups.

Using Dahmani’s boundary construction for relatively hyperbolic groups together with the uniform convergence group characterization of hyperbolic TDLC groups due to Carette‑Dreesen, the authors prove Theorem 1.2: if all vertex groups are hyperbolic TDLC, each edge group embeds quasiconvexly into its adjacent vertex groups, and the global action is acylindrical, then the fundamental group (G) is itself a hyperbolic TDLC group and each vertex group is quasiconvex in (G). The proof builds an explicit candidate boundary (X) by gluing the Gromov boundaries of the vertex and edge groups, endowing (X) with a natural topology, and showing that the induced action of (G) on (X) is a uniform convergence action.

The paper then studies the height of subgroups in this setting. Extending the GMRS and Mitra results, Theorem 1.3 shows that for a hyperbolic TDLC group splitting as an amalgamated product or HNN extension, an edge subgroup (C) is quasiconvex in the whole group if and only if its height (the maximal number of essentially distinct conjugates intersecting non‑trivially) is finite. This provides a new proof even for discrete groups and demonstrates that the height‑quasiconvexity equivalence survives the passage to TDLC groups.

The authors introduce a notion of local quasiconvexity for TDLC groups: a hyperbolic TDLC group (G) is locally quasiconvex if every open, compactly generated subgroup is quasiconvex. They prove that such groups satisfy the Howson property (intersection of two finitely generated subgroups is finitely generated) and are coherent (open compactly generated subgroups are compactly presented).

The main combination result, Theorem 1.4, states that if a finite graph of TDLC groups satisfies: (i) vertex groups are locally quasiconvex hyperbolic TDLC, (ii) edge groups are quasiconvex in adjacent vertices, (iii) the global action is acylindrical, and (iv) the intersection of a compactly generated subgroup of (G) with any conjugate of an edge group is again compactly generated, then the fundamental group (G) is a locally quasiconvex hyperbolic TDLC group. Conversely, if (G) is locally quasiconvex under the same hypotheses (except (iv)), then each vertex group is locally quasiconvex in (G). The theorem subsumes several earlier combination theorems for discrete groups and yields corollaries such as: when edge groups are compact, local quasiconvexity of the whole group is equivalent to that of the vertex groups; groups quasi‑isometric to locally finite trees are locally quasiconvex hyperbolic (hence (\mathrm{SL}(2,\mathbb{Q}_p)) is locally quasiconvex).

The final part of the paper addresses Cannon‑Thurston maps for extensions of hyperbolic TDLC groups. Building on Mosher’s quasi‑isometric section technique, the authors construct a quasi‑isometric section for any short exact sequence of compactly generated hyperbolic TDLC groups
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