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In this paper, we introduce a new class of non-amenable groups denoted by   $\textbf {NC}_1 \cap \textbf {Quot}(\mathcal C_{rss})$   which give rise to  prime  von Neumann algebras. This means that, for every   $\Gamma \in \textbf {NC}_1 \cap \textbf {Quot}(\mathcal {C}_{rss}),$   its group von Neumann algebra   $L(\Gamma )$   cannot be decomposed as a tensor product of diffuse von Neumann algebras. We show   $\textbf {NC}_1 \cap \textbf {Quot}(\mathcal C_{rss})$   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.

Primeness Results for von Neumann Algebras Associated with Surface Braid Groups