Primeness results for von Neumann algebras associated with surface braid groups

TL;DR

This paper introduces a new class of non-amenable groups that produce prime von Neumann algebras, expanding understanding of algebraic structures associated with complex surface braid groups and related groups.

Contribution

It defines the class ${\bf NC}_1 \cap {\bf Quot}(\mathcal C_{rss})$ and proves that their group von Neumann algebras are prime, including many well-studied groups such as surface braid groups and mapping class groups.

Findings

Groups in the class produce prime von Neumann algebras.

Includes many groups from surface and mapping class groups.

Shows the class is large and encompasses various important 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.