The first $\ell^2$-Betti number and groups acting on trees

TL;DR

This paper extends previous work on the first  ext{-Betti number} to a broader class of groups acting on trees, providing new insights into their algebraic and geometric properties.

Contribution

It generalizes existing results on the first  ext{-Betti number} to quotients of groups acting on trees with specific free actions on edges.

Findings

Extended the class of groups for which the first  ext{-Betti number} can be computed.

Provided new formulas relating group actions on trees to their  ext{-Betti number}.

Enhanced understanding of the structure of groups acting on trees.

Abstract

We generalise results of Thomas, Allcock, Thom-Petersen, and Kar-Niblo to the first $\ell^2$-Betti number of quotients of certain groups acting on trees by subgroups with free actions on the edge sets of the graphs.