Group cocycles and the ring of affiliated operators

TL;DR

This paper investigates group cocycles with values in l^2(G) and explores the structure of the ring of affiliated operators, providing new insights into group cohomology and subgroup properties related to the first l^2-Betti number.

Contribution

It clarifies properties of the first cohomology with l^2(G) coefficients and establishes new results on free subgroups and subgroup structure for groups with positive first l^2-Betti number.

Findings

Characterization of first cohomology properties

Existence of free subgroups in certain groups

Examples of groups satisfying the conditions

Abstract

In this article we study cocycles of discrete countable groups with values in l^2(G) and the ring of affiliated operators UG. We clarify properties of the first cohomology of a group G with coefficients in l^2(G) and answer several questions from [CTV]. Moreover, we obtain strong results about the existence of free subgroups and the subgroup structure, provided the group has a positive first l^2-Betti number. We give numerous applications and examples of groups which satisfy our assumptions.