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 ${\mathbf{NC}}_1\cap \mathbf{Quot}({\mathcal{C}}_{rss})$ which give rise to $\text{prime}$ von Neumann algebras. This means that for every $\Gamma \in {\mathbf{NC}}_1\cap \mathbf{Quot}({\mathcal{C}}_{rss})$ its group von Neumann algebra $L(\Gamma )$ cannot be decomposed as a tensor product of diffuse von Neumann algebras. We show ${\mathbf{NC}}_1\cap \mathbf{Quot}({\mathcal{C}}_{rss})$ 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.

arXiv comment32 pages

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Cited by in corpus (3)

• OE and W* superrigidity results for actions by surface braid groups
• Some Applications of Group Theoretic Rips Constructions to the Classification of von Neumann Algebras
• Tensor product decompositions of II${}_1$ factors arising from extensions of amalgamated free product groups