---
title: Graph of Locally Quasi-Convex Hyperbolic Groups
url: https://www.emergentmind.com/topics/graph-of-locally-quasi-convex-hyperbolic-groups
type: topic
---

# Graph of Locally Quasi-Convex Hyperbolic Groups

A graph of locally quasi-convex hyperbolic groups is a combinatorial and geometric structure in which each vertex corresponds to a word-hyperbolic group possessing the property of local quasi-convexity, and each edge subgroup is infinite cyclic and quasi-convex as embedded in its adjacent vertex groups. The study of such structures occupies a central position in geometric group theory, blending Bass–Serre theory, Gromov’s hyperbolicity, quasi-convexity, and boundary theory, with deep connections to questions of subgroup structure, local-to-global phenomena, and algorithmic properties.

## 1. Definitions and Fundamental Structure

Let \( Y \) be a finite connected graph, and to each vertex \( v \in V(Y) \) assign a word-hyperbolic group \( G_v \), requiring that each \( G_v \) is locally quasi-convex—that is, every finitely generated subgroup of \( G_v \) is quasi-convex. To each edge \( e \) associate a subgroup \( E_e \cong \mathbb{Z} \) that embeds as a quasi-convex (hence undistorted) subgroup into both incident vertex groups \( G_v \), \( G_w \).

The resulting structure is formalized as a graph of groups, and its fundamental group, denoted \( G = \pi_1(G(Y)) \), is assembled via the usual Bass–Serre construction. In particular, the edge groups’ quasi-convexity and local quasi-convexity of vertex groups underlie all major combination theorems.

## 2. Combination Theorems and Hyperbolicity

The core result for graphs of locally quasi-convex hyperbolic groups is a strong combination theorem. Tomar’s Theorem 1.3 asserts that if each vertex group acts as a convergence group on its Gromov boundary and every edge group is dynamically quasi-convex and forms an almost malnormal family in the vertex groups, then the fundamental group is a convergence group on an explicit compactum, i.e., acts as a convergence group on a compact metrizable space [2104.08843].

For the absolute (non-relative) case where every vertex group is word-hyperbolic and locally quasi-convex, and every edge group is infinite cyclic, quasi-convex, and almost malnormal among the incident edges, the combination theorem specializes to:

- **Hyperbolicity of \( G \)**: \( G \) is itself word-hyperbolic.
- **Local Quasi-Convexity**: \( G \) is locally quasi-convex, i.e., every finitely generated subgroup is quasi-convex.
- **Quasi-Convexity of Subgroups**: Subgroup \( H \leq G \) is quasi-convex in \( G \) if and only if each intersection \( H \cap G_v \) is quasi-convex in \( G_v \) [1211.1993].

These results rely on the Bestvina–Feighn combination theorem and fine-graph techniques, with quasiconvex ℤ-edge subgroups guaranteeing 2-acylindricity of the Bass–Serre tree action [2104.08843, 1211.1993].

## 3. Construction and Classification of Boundaries

The Gromov boundary \( \partial G \) of \( G \) admits an explicit model built via “gluing” the boundaries \( \partial G_v \) of the vertex groups along 2-point limit sets corresponding to images of the cyclic edge groups. Let

\[
M := \left[ \bigsqcup_{g \in G, v \in V(Y)} \{g\} \times \partial G_v \right] \sqcup \partial T \,/\,{\sim}
\]

where \( T \) is the Bass–Serre tree, with identification:

\[
(gG_v, \lambda_{e,v}(\xi)) \sim (gG_w, \lambda_{e,w}(\xi))
\]

for each edge \( e = (v,w) \), \( g \in G \), \( \xi \in \Lambda(E_e) \). The resulting compactum supports a convergence action of \( G \), and by Yaman’s theorem, \( M \) is equivariantly homeomorphic to \( \partial G \) [2104.08843].

The homeomorphism type of \( \partial G \) is completely determined by:

| Data            | Description                           | Role                             |
|-----------------|---------------------------------------|----------------------------------|
| (i)             | Underlying graph \( Y \)              | Topological skeleton             |
| (ii)            | Boundaries \( \partial G_v \)         | Structure of the pieces          |
| (iii)           | Images \( \Lambda(E_e) \subset \partial G_v \) | Gluing loci for edge groups |

Two such graphs of groups with matching data (i)–(iii) yield equivariantly homeomorphic boundaries [2104.08843].

## 4. Quasi-Convexity: Permanence and Algorithmic Aspects

Natural subgroups arising from subgraphs or single vertex groups are quasi-convex in \( G \). Specifically:

- Any finitely generated subgroup carried entirely by a vertex group \( G_v \) is quasi-convex in \( G \).
- Any subgroup arising as the fundamental group of a subgraph is quasi-convex [2104.08843, 1211.1993].

This quasi-convexity permanence enables the algorithmic construction of Stallings graphs for all finitely generated subgroups. In the context of locally quasi-convex hyperbolic groups, the Stallings folding procedure, based on automata-theoretic techniques, produces a finite, canonical labeled graph \( \Gamma(H) \) for any finitely generated subgroup \( H \), allowing effective solution of subgroup membership, intersection, conjugacy, and almost malnormality problems [1408.1917]. Key algorithmic implications include:

- Decidability of quasi-convexity via construction of \( \Gamma(H) \).
- Functoriality of Stallings graphs under inclusion and intersection.
- Complete computable “atlas” of finitely generated subgroups.

## 5. Boundary Theory and Rigidity

The boundary theory for graphs of locally quasi-convex hyperbolic groups is rigid. The “gluing” model ensures that the homeomorphism type of \( \partial G \) is a complete invariant for the combination, governed solely by the topological data of the graph and boundaries, along with the embeddings of the cyclic edge groups.

Criterion: Two graphs of hyperbolic groups with cyclic quasi-convex edges whose gluing data agree via a graph isomorphism and boundary homeomorphisms induce an equivariant homeomorphism of boundaries [2104.08843]. This yields a classification up to equivariant homeomorphism, depending only on local and combinatorial data.

## 6. Relativity, Combination, and Finiteness Theorems

In the context of relatively hyperbolic groups, analogous statements hold. If \( G \) splits as a graph of groups with each vertex group relatively hyperbolic and each edge group parabolic and relatively quasi-convex, then \( G \) is hyperbolic relative to a canonical collection of “parabolic tree” stabilizers. The theory extends to relative quasi-convexity, local relative quasi-convexity, and provides if-and-only-if criteria for both global (relative) hyperbolicity and quasi-convexity of subgroups [1211.1993, 2403.17686].

Finiteness results: For a finitely generated torsion-free locally relatively quasi-convex relatively hyperbolic group, there are only finitely many isomorphism classes of n-generated subgroups, each represented by a folded carrier graph [2403.17686].

## 7. Applications and Examples

Explicit instances range from free products with cyclic amalgamation (e.g., \( F(a,b) *_{\langle c \rangle} \)) to fundamental groups of graphs of surface or free groups. For Kleinian groups, all finitely generated subgroups are relatively quasi-convex, and the structure of their graphs encodes fine combinatorial and geometric information [2403.17686].

Graph-theoretic and automata-theoretic descriptions undergird algorithmic applications, including computation of quasi-convexity constants, subgroup intersection, conjugacy, and enumeration of subgroup types. The interplay between topological invariants, group-theoretic properties, and algorithmic models such as Stallings graphs crystallizes the subject’s modern scope [1408.1917].

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**Key references:**
- "Boundaries of graphs of relatively hyperbolic groups with cyclic edge groups" [2104.08843]
- "Quasiconvexity and relatively hyperbolic groups that split" [1211.1993]
- "Stallings graphs for quasi-convex subgroups" [1408.1917]
- "Foldings in relatively hyperbolic groups" [2403.17686]

Source: https://www.emergentmind.com/topics/graph-of-locally-quasi-convex-hyperbolic-groups