Group extensions with infinite conjugacy classes

TL;DR

This paper characterizes when groups formed by various types of extensions have infinite conjugacy classes (icc), linking group structure to properties of associated von Neumann algebras and providing comprehensive criteria for different extension types.

Contribution

It provides a complete characterization of the icc property for groups constructed via direct, semi-direct, wreath, and general extensions, including special cases with additional hypotheses.

Findings

Characterization of icc property for direct product groups

Criteria for semi-direct product groups to be icc

Conditions under which wreath product groups are icc

Abstract

We characterize the group property of being with infinite conjugacy classes (or icc, i.e. infinite and of which all conjugacy classes except {1} are infinite) for groups which are defined by an extension of groups. We give characterizations for all different kinds of extension: direct product, semi-direct product, wreath products and general extension. We also give many particular results when the groups involved verify some additional hypothesis. The icc property is correlated to the Theory of Von Neumann algebras since a necessary and sufficient condition for the Von Neumann algebra of a group \Gamma to be a factor of type II - 1, is that \Gamma be icc. Our approach applies in full generality in the study of icc property since any group either decomposes as an extension of groups or is simple, and in the latter case icc property becomes trivially equivalent to being infinite.