C$^*$-simple groups: amalgamated free products, HNN extensions, and fundamental groups of 3-manifolds

Pierre de la Harpe, Jean-Philippe Préaux

Published 2009-09-18, updated 2011-06-26Version 2

We establish sufficient conditions for the C$^*$-simplicity of two classes of groups. The first class is that of groups acting on trees, such as amalgamated free products, HNN-extensions, and their normal subgroups; for example normal subgroups of Baumslag-Solitar groups. The second class is that of fundamental groups of compact 3-manifolds, related to the first class by their Kneser-Milnor and JSJ-decompositions. Much of our analysis deals with conditions on an action of a group $\Gamma$ on a tree $T$ which imply the following three properties: abundance of hyperbolic elements, better called strong hyperbolicity, minimality, both on the tree $T$ and on its boundary $\partial T$, and faithfulness in a strong sense. An important step in this analysis is to identify automorphism of $T$ which are \emph{slender}, namely such that their fixed-point sets in $\partial T$ are nowhere dense for the shadow topology.