Graph of Locally Quasi-Convex Hyperbolic Groups

• The paper demonstrates that the fundamental group of a graph of locally quasi-convex hyperbolic groups is hyperbolic and retains local quasi-convexity.
• It leverages Bass–Serre theory and Stallings graph techniques to analyze subgroup intersections and solve membership problems efficiently.
• The study provides a boundary construction that glues together vertex group boundaries, serving as a complete invariant for group classification.

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 (Tomar, 2021).

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 $G_v$ (Bigdely et al., 2012).

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 (Tomar, 2021, Bigdely et al., 2012).

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 $\partial G$ (Tomar, 2021).

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 (Tomar, 2021).

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 (Tomar, 2021, Bigdely et al., 2012).

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 (Kharlampovich et al., 2014). 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 (Tomar, 2021). 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 (Bigdely et al., 2012, Weidmann et al., 2024).

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 (Weidmann et al., 2024).

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 (Weidmann et al., 2024).

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 (Kharlampovich et al., 2014).

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Key references:

• "Boundaries of graphs of relatively hyperbolic groups with cyclic edge groups" (Tomar, 2021)
• "Quasiconvexity and relatively hyperbolic groups that split" (Bigdely et al., 2012)
• "Stallings graphs for quasi-convex subgroups" (Kharlampovich et al., 2014)
• "Foldings in relatively hyperbolic groups" (Weidmann et al., 2024)