Pullbacks and intersections in categories of graphs of groups

We develop a categorical framework for studying graphs of groups and their morphisms, with emphasis on pullbacks. More precisely, building on classical work by Serre and Bass, we give an explicit construction of the so-called $\mathbb{A}$-product of two morphisms into a graph of groups $\mathbb{A}$ – a graph of groups which, within the appropriate categorical setting, captures the intersection of subgroups of the fundamental group of $\mathbb{A}$. We show that, in the category of pointed graphs of groups, pullbacks always exist and correspond precisely to pointed $\mathbb{A}$-products. In contrast, pullbacks do not always exist in the category of unpointed graphs of groups. However, when they do exist, and we show that it is the case, in particular, under certain acylindricity conditions, they are again closely related to $\mathbb{A}$-products. We trace, all along, the parallels with Stallings’ classical theory of graph immersions and coverings, in relation to the study of the subgroups of free groups. Our results are useful for studying intersections of subgroups of groups that arise as fundamental groups of graphs of groups. As an example, we carry out an explicit computation of a pullback which results in a classification of the Baumslag–Solitar groups with the finitely generated intersection property.

💡 Research Summary

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The paper develops a categorical framework for studying graphs of groups and their morphisms, focusing on the existence and construction of pullbacks, which model subgroup intersections. Building on Bass–Serre theory, the authors introduce two categories: GrGp, whose objects are (unpointed) graphs of groups and morphisms are taken up to a relation denoted «; and GrGp°, whose objects are connected pointed graphs of groups with morphisms taken up to a relation denoted „.

The central construction is the 𝔄‑product (written B ̂_𝔄 C) associated to two morphisms μ_B : B → 𝔄 and μ_C : C → 𝔄. The underlying graph of the product is the ordinary pullback of the underlying graphs of B and C. For each vertex or edge of this pullback, the authors consider the double cosets formed by the images of the corresponding vertex or edge groups in the ambient group of 𝔄 under μ_B and μ_C. These double cosets become the vertex and edge groups of the product, equipped with appropriate “twisted” pullback maps. Natural projection morphisms ρ_B and ρ_C from the product to B and C are defined, and the whole construction is shown to be independent of auxiliary parameters up to isomorphism in the appropriate category.

Theorem A (Theorem 3.14) proves that in the pointed category GrGp°, pullbacks always exist. Moreover, the pullback of two morphisms is precisely their pointed 𝔄‑product. The proof hinges on a lifting lemma: given a diagram D → B and D → C whose composites with μ_B and μ_C are equivalent, there exists a morphism D → B ̂_𝔄 C making the diagram commute up to the chosen equivalence. All possible lifts are described via the double‑coset structure. When the original morphisms are immersions (locally injective), the pullback coincides with a single connected component of the product, mirroring the classical Stallings pullback for free groups.

In contrast, Theorem B (Example 3.19) shows that pullbacks need not exist in the unpointed category GrGp. The failure is illustrated by a concrete example where the required universal property cannot be satisfied because the images of edge groups do not align appropriately. However, under an acylindricity hypothesis—roughly, the action of the ambient group on the Bass–Serre tree has uniformly bounded stabilizers—pullbacks do exist (Theorem E). In this situation the pullback embeds as a subgraph of the 𝔄‑product, and the universal property holds within that subgraph.

Further results (Theorems C and D) relate the connected components of the 𝔄‑product to intersections of conjugates of the fundamental groups of B and C inside π₁(𝔄). Each component corresponds to a double coset class, and the fundamental group of that component is precisely the intersection of the corresponding conjugates. This provides a direct analogue of the Stallings description of subgroup intersections via pullbacks of finite graphs.

The authors apply the theory to the standard graph of groups of a Baumslag–Solitar group BS(p,m,n) (with |p|,|m|,|n|>1). By constructing two explicit immersions of pointed graphs of groups into this standard graph, they compute the 𝔄‑product and identify a connected component that serves as the pullback. The resulting pullback describes two finitely generated subgroups whose intersection is not finitely generated, reproducing a known counter‑example to the finitely generated intersection property (f.g.i.p.) in BS groups (originally due to Paraman­tzoglou).

Overall, the paper achieves three major contributions:

1. Existence and explicit construction of pullbacks in the pointed category of graphs of groups via the 𝔄‑product, extending the classical Stallings pullback from free groups to the much broader Bass–Serre setting.

2. Identification of precise conditions (acylindricity) under which pullbacks exist in the unpointed category, together with a description of how such pullbacks sit inside the 𝔄‑product.

3. Concrete connection between categorical pullbacks and subgroup intersections, including a detailed analysis of double‑coset structures and an explicit example in Baumslag–Solitar groups that demonstrates the failure of f.g.i.p.

These results open the way for algorithmic approaches to subgroup intersection problems in groups that are fundamental groups of graphs of groups, and they suggest further extensions to relatively hyperbolic groups, virtually free groups, and other classes where Bass–Serre theory provides a natural combinatorial model.