Euler Characteristics on virtually free products

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 formulas are obtained from a general result which applies to the rank gradient and the first L²−Betti number of a finitely generated group.

KEYWORDS:

• Euler characteristics
• first L²−Betti number
• rank gradient
• residually finite groups
• virtually torsion free groups

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20E06, 20E26

Acknowledgments

I would like to thank Mihalis Sykiotis for helpful suggestions and discussions. I also thank Denis Osin for useful comments on an earlier draft of this paper. Finally, I also thank the referee whose comments and suggestions improved this paper as well as for pointing out a mistake in the statement of Lemma 2.1 in the previous version.