Primeness Results for von Neumann Algebras Associated with Surface Braid Groups

Abstract

In this paper we introduce a new class of non-amenable groups denoted by ${\bf NC}_1 \cap {\bf Quot}(\mathcal C_{rss})$ which give rise to $\textit{prime}$ von Neumann algebras. This means that for every $\Gamma \in {\bf NC}_1 \cap {\bf Quot}(\mathcal C_{rss})$ its group von Neumann algebra $L(\Gamma)$ cannot be decomposed as a tensor product of diffuse von Neumann algebras. We show ${\bf NC}_1 \cap {\bf Quot}(\mathcal C_{rss})$ is fairly large as it contains many examples of groups intensively studied in various areas of mathematics, notably: all infinite central quotients of pure surface braid groups; all mapping class groups of (punctured) surfaces of genus $0,1,2$; most Torelli groups and Johnson kernels of (punctured) surfaces of genus $0,1,2$; and, all groups hyperbolic relative to finite families of residually finite, exact, infinite, proper subgroups.