The works of F. Murray and J. von Neumann outlined a procedure to associate a von Neumann algebra to a group. Since then, an active area of research investigates which structural aspects of the group are detectable in its von Neumann algebra.  The difficulty of this problem is best illustrated by Conne's landmark result which states all countable ICC amenable groups give rise to isomorphic objects.  In essence, standard group invariants alone are typically too weak to be detectable the resulting von Neumann algebra.  However, when the group is non-amenable the situation may be strikingly different.

This talk surveys advances made in this area, with an emphasis on the results stemming from Popa's deformation/rigidity theory.  I present several instances where elementary structural features of a group, such as the direct product, can be recovered from the algebra.  We will then discuss recent progress made by Ben Hayes, Dan Hoff, Thomas Sinclair and myself on the analysis of s-malleable deformations, in the sense of Popa,  of tracial von Neumann algebras and its relationship to $\ell^2$ cohomology theory of groups.    Finally, we will detail the applications of our work which may resolve open conjectures of Peterson and Thom for von Neumann algebra of the free group $\mathbb{F}_2$.