We characterize the group property of being with infinite conjugacy classes (or icc, i.e. infinite and of which all conjugacy classes except $\left\{1\right\}$ are infinite) for groups which are extensions of groups. We prove a general result for extensions of groups, then deduce characterizations in semi-direct products, wreath products, finite extensions, among others examples we also deduce a characterization for amalgamated products and HNN extensions. 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 discrete group $\Gamma $ to be a factor of type $I{I}_{1}$, is that $\Gamma $ be icc. Our approach applies in full generality to the study of icc property since any group that does not split as an extension is simple, and in such case icc property becomes equivalent to being infinite.

@article{CML_2013__5_1_73_0, author = {Pr\'eaux, Jean-Philippe}, title = {Group Extensions with Infinite Conjugacy Classes}, journal = {Confluentes Mathematici}, publisher = {Institut Camille Jordan}, volume = {5}, number = {1}, year = {2013}, pages = {73-92}, doi = {10.5802/cml.3}, mrnumber = {3143612}, language = {en}, url = {http://www.numdam.org/item/CML_2013__5_1_73_0} }

Préaux, Jean-Philippe. Group Extensions with Infinite Conjugacy Classes. Confluentes Mathematici, Volume 5 (2013) no. 1, pp. 73-92. doi : 10.5802/cml.3. http://www.numdam.org/item/CML_2013__5_1_73_0/

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