Groupes fondamentaux des varietes de dimension 3 et algebres d'operateurs

We provide a geometric characterization of manifolds of dimension 3 with fundamental groups of which all conjugacy classes except 1 are infinite, namely of which the von Neumann algebras are factors of type $I{I}_1$: they are essentially the 3-manifolds with infinite fundamental groups on which there does not exist any Seifert fibration. Otherwise said and more precisely, let $M$ be a compact connected 3-manifold and let $\mathrm{\Gamma }$ be its fundamental group, supposed to be infinite and with at least one finite conjugacy class besides 1. If $M$ is orientable, then $\mathrm{\Gamma }$ is the fundamental group of a Seifert manifold; if $M$ is not orientable, then $\mathrm{\Gamma }$ is the fundamental group of a Seifert manifold modulo $\mathbb{P}$ in the sense of Heil and Whitten \cite{HeWh–94}. We make heavy use of results on 3-manifolds, as well classical results (as can be found in the books of Hempel, Jaco, and Shalen), as more recent ones (solution of the Seifert fibred space conjecture).