Euler characteristics on virtually free products

TL;DR

This paper introduces Euler characteristics for certain groups and demonstrates their behavior in amalgamated free products and HNN extensions, linking these to rank gradient and $L^2$-Betti numbers.

Contribution

It defines Euler characteristics for residually finite and virtually torsion free groups and establishes formulas for their behavior in specific group constructions.

Findings

Euler characteristics satisfy specific formulas in amalgamated free products and HNN extensions.

Formulas relate Euler characteristics to rank gradient and first $L^2$-Betti number.

Results apply to residually finite and virtually torsion free groups.

Abstract

We define Euler characteristics on classes of residually finite and virtually torsion free groups and we show that they satisfy certain formulas in the case of amalgamated free products and HNN extensions over finite subgroups. These forumlas are obtained from a general result which applies to the rank gradient and the first $L^2$-Betti number of a finitely generated group.