Splittings of Non-Finitely Generated Groups

TL;DR

This paper extends the theory of group splittings to non-finitely generated groups by utilizing Bass-Serre trees, removing the need for local finiteness assumptions inherent in Cayley graph methods.

Contribution

It generalizes the intersection theory of group splittings from finitely generated groups to a broader class without local finiteness constraints.

Findings

Reformulation of intersection theory using Bass-Serre trees.

Removal of finite generation assumptions in splitting analysis.

Enhanced understanding of group actions on non-locally finite trees.

Abstract

In geometric group theory one uses group actions on spaces to gain information about groups. One natural space to use is the Cayley graph of a group. The Cayley graph arguments that one encounters tend to require local finiteness, and hence finite generation of the group. In this paper, I take the theory of intersections of splittings of finitely generated groups (as developed by Scott, Scott-Swarup, and Niblo-Sageev-Scott-Swarup), and rework it to remove finite generation assumptions. Whereas the aforementioned authors relied on the local finiteness of the Cayley graph, I capitalize on the Bass-Serre trees for the splittings.