C${}^{\ast }$-simple groups: amalgamated free products, HNN extensions, and fundamental groups of 3-manifolds

Abstract

We establish sufficient conditions for the C${}^{\ast }$-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.

43 pages

References in corpus (3)

• Notes on Perelman's papers
• Completion of the Proof of the Geometrization Conjecture
• Characterization of linear groups whose reduced C*-algebras are simple

Cited by in corpus (14)

• Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces
• C*-simplicity and the unique trace property for discrete groups
• An intrinsic characterization of C*-simplicity
• C*-simplicity of free products with amalgamation and radical classes of groups
• C*-algebras associated to graphs of groups
• $C^{\ast }$-simple groups without free subgroups
• C*-irreducibility for reduced twisted group C*-algebras
• Limits of Baumslag-Solitar groups and dimension estimates in the space of marked groups
• Two Families of Examples of Groups Acting on Trees with Nontrivial Quasi-Kernels
• Presentability by products for some classes of groups
• On maximal ideals in certain reduced twisted C*-crossed products
• A characterization of high transitivity for groups acting on trees
• $C^{\ast }$-simplicity for groups with non-elementary convergence group actions
• Simplicity of crossed products of the actions of totally disconnected locally compact groups on their boundaries