Infinite graph product of groups I: Geometry of the extension graph

We introduce the extension graph of graph product of groups and study its geometry. This enables us to study properties of graph product by exploiting large scale geometry of its defining graph. In particular, we show that the extension graph is isomorphic to the crossing graph of a canonical quasi-median graph and exhibits the same phenomenon about asymptotic dimension as quasi-trees of metric spaces studied by Bestvina-Bromberg-Fujiwara. As an application of the extension graph, we prove relative hyperbolicity of graph-wreath product. This provides a new construction of relatively hyperbolic groups.

💡 Research Summary

This paper introduces the extension graph Γᵉ for graph products of groups and develops a systematic geometric framework that connects the large‑scale geometry of the defining simplicial graph Γ with the algebraic and coarse properties of the graph product Γ G.

The authors begin by recalling the construction of a graph product: given a simplicial graph Γ and a collection of vertex groups {G_v | v∈V(Γ)}, the graph product Γ G is the quotient of the free product of the G_v’s by the relations that elements of adjacent vertex groups commute. While this construction generalizes direct products (Γ complete) and free products (Γ edgeless), most existing results focus on finite Γ. The paper’s central goal is to understand how the geometry of an arbitrary (possibly infinite) defining graph influences the geometry of the resulting group.

Definition of the extension graph.
In Definition 3.1 the extension graph Γᵉ is defined as follows: its vertices are the conjugates of non‑trivial elements of the vertex groups inside Γ G, and two vertices are joined by an edge precisely when the corresponding group elements commute. This definition coincides with the Kim–Koberda extension graph for right‑angled Artin groups (RAAGs) but works for any graph product, even when the vertex groups are infinite. The construction is natural because Γ G acts on Γᵉ by conjugation, making Γᵉ a Γ G‑invariant graph that records the commutation pattern of all conjugates.

Relation to quasi‑median geometry.
Section 3.2 proves that Γᵉ is isomorphic to the crossing graph of a canonical quasi‑median graph X(Γ,G) associated to the graph product. The crossing graph records intersections of hyperplanes in X; its vertices are hyperplanes, and edges correspond to intersecting hyperplanes. This identification generalizes the known correspondence for RAAGs and allows the authors to import the rich combinatorial and metric theory of quasi‑median graphs (hyperplane structure, contact graph, etc.) into the study of extension graphs. Consequently, many geometric properties of Γᵉ can be read off from the well‑understood geometry of X(Γ,G).

Asymptotic dimension.
Theorem 1.1 is a central quantitative result: if Γ is connected, has girth > 20, and its asymptotic dimension satisfies asdim(Γ)=n, then the extension graph satisfies asdim(Γᵉ) ≤ n+1. The proof proceeds by constructing an auxiliary “coned‑off” version of Γᵉ, showing that Γᵉ can be viewed as a tree‑like amalgamation of copies of Γ. The argument mirrors the Bestvina–Bromberg–Fujiwara analysis of quasi‑trees of metric spaces, where gluing together uniformly bounded pieces along a tree adds at most one to the asymptotic dimension. The girth condition guarantees that short cycles do not create unexpected shortcuts that could increase the dimension.

Hyperbolicity, fineness, and Bowditch tightness.
Theorem 1.2 contains three parts:

1. Γ is Gromov‑hyperbolic if and only if Γᵉ is hyperbolic. This equivalence holds under the same girth > 20 hypothesis and shows that the extension graph faithfully reflects the large‑scale negative curvature of the defining graph.

2. If Γ is uniformly fine (every edge lies in uniformly boundedly many simple cycles) and hyperbolic, then Γᵉ satisfies Bowditch’s tightness condition. Tightness is a strengthening of hyperbolicity that implies acylindrical actions and finiteness of asymptotic dimension; thus Γᵉ inherits strong geometric control from Γ.

3. If Γ is fine and all vertex groups are finite, then Γᵉ is also fine. Fineness is a combinatorial analogue of local finiteness for infinite graphs and is crucial in the study of curve complexes and relatively hyperbolic groups. This result provides a new source of fine hyperbolic graphs beyond the classical examples (curve complexes, coned‑off Cayley graphs).

Application to graph wreath products.
The paper then applies the developed theory to a construction called the graph wreath product. Given a group G acting on a simplicial graph Γ and a finite group H, one defines a new graph product where each vertex group is H, and then forms the semidirect product (Γ H) ⋊ G. This construction interpolates between the classical wreath product H ≀ G (when Γ is a complete graph on the underlying set of G) and a free product of copies of H indexed by the vertices of Γ (when Γ has no edges).

Theorem 1.3 states that if Γ is a fine hyperbolic graph with girth > 20, G is finitely generated and acts on Γ with finite edge stabilizers and finitely generated vertex stabilizers, then the resulting graph wreath product is relatively hyperbolic relative to a natural collection of subgroups (the stabilizers of vertices and the “cone‑off” subgroups). Corollary 1.4 strengthens this: when Γ is locally finite, hyperbolic, and G acts properly and cocompactly, the graph wreath product is actually hyperbolic (not merely relatively hyperbolic). The girth > 20 and the condition |H| ≠ 2 are essential because the presence of a Z₂ subgroup would introduce a flat plane, contradicting hyperbolicity.

Technical remarks and outlook.
The girth > 20 hypothesis appears repeatedly; it prevents short cycles from creating unwanted commutation relations that could break the tree‑like decomposition of Γᵉ. The authors note that one can always increase girth by barycentric subdivision, so the condition is not restrictive in practice. The paper also discusses how the extension graph provides a new perspective on the crossing and contact graphs of quasi‑median spaces, complementing recent work that studies these graphs via quasi‑median Cayley graphs.

In summary, the paper establishes the extension graph as a powerful geometric invariant of graph products. By proving that Γᵉ is isomorphic to the crossing graph of a canonical quasi‑median space, the authors transfer a host of geometric properties—such as asymptotic dimension bounds, hyperbolicity, fineness, and Bowditch tightness—from the defining graph Γ to the group‑level object Γᵉ. The application to graph wreath products yields a novel method for constructing relatively hyperbolic and hyperbolic groups, enriching the landscape of groups with controlled large‑scale geometry. Future work, hinted at by the authors, will exploit the extension graph to investigate analytic properties (e.g., rapid decay, L²‑cohomology) and boundary phenomena for graph products.