Pullbacks and intersections in categories of graphs of groups

TL;DR

This paper develops a categorical framework for graphs of groups, focusing on pullbacks and intersections, and relates these to classical theories and subgroup properties, with explicit computations for Baumslag--Solitar groups.

Contribution

It introduces an explicit construction of the $ ext{A}$-product for morphisms into a graph of groups and characterizes pullbacks in pointed and unpointed categories, linking to subgroup intersection theory.

Findings

Pullbacks always exist in the category of pointed graphs of groups.

Pullbacks correspond to pointed $ ext{A}$-products in pointed categories.

Explicit computation classifies Baumslag--Solitar groups with the finitely generated intersection property.

Abstract

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.