Classification of tensor decompositions for II$_1$ factors

In the mid thirties Murray and von Neumann found a natural way to associate a von Neumann algebra $L(\Gamma)$ to any countable discrete group $\Gamma$. Classifying $L(\Gamma)$ in term of $\Gamma$ is a notoriously complex problem as in general the initial data tends to be lost in the von Neumann algebraic regime. An important problem in the theory of von Neumann algebras is to completely describe all possible tensor decompositions of a given group von Neumann algebra $L(\Gamma)$. In this direction the main goal is to investigate how exactly a tensor decomposition of $L(\Gamma)$ relates to the underlying group $\Gamma$. In this dissertation we introduce several new classes of groups $\Gamma$ for which all tensor decompositions of $L(\Gamma)$ are parametrized by the canonical direct product decompositions of $\Gamma$. Specifically, we show that whenever $L(\Gamma)\cong M_1\bar\otimes M_2$ where $M_i$ are any diffuse von Neumann algebras then there exists a non-canonical direct product decomposition $\Gamma=\Gamma_1\times\Gamma_2$ such that up to amplifications we have that $M_1\cong L(\Gamma_1)$ and $M_2\cong L(\Gamma_2)$. Our class include large classes of icc (infinite conjugacy class) amalgamated free products and wreath product groups. In addition we obtain similar classifications of tensor decompositions for the von Neumann algebras associated with the $T_0$ and $T_1$ group functors introduced by McDuff in $1969$.