A simple construction of infinite finitely generated torsion groups

The goal of this note is to provide yet another proof of the following theorem of Golod: there exists an infinite finitely generated group $G$ such that every element of $G$ has finite order. Our proof is based on the Nielsen-Schreier index formula and is simple enough to be included in a standard group theory course.Comment: The idea of the proof is not really original. The paper is written for expository purpose

Amenable actions, free products and a fixed point property

We investigate the class of groups admitting an action on a set with an invariant mean. It turns out that many free products admit such an action. We give a complete characterisation of such free products in terms of a strong fixed point property.Comment: 12 page