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 intersection numbers of splittings of finitely generated groups (as developed by Scott, Swarup, Niblo and Sageev), and rework it to remove finite generation assumptions. I show that when working with splittings, instead of using the Cayley graph, one can use Bass–Serre trees.
"Splittings of non-finitely generated groups." Algebr. Geom. Topol. 12 (1) 511 - 563, 2012. https://doi.org/10.2140/agt.2012.12.511