The works of F. Murray and J. von Neumann outlined a natural method to associate a von Neumann algebra to a group. Since then, an active area of research seeks to investigate which structural aspects of the group extend to its von Neumann algebra.  The difficulty of this problem is best illustrated by Conne's landmark result which states all ICC amenable groups give rise to isomorphic von Neumann algebras.  In essence, standard group invariants are not typically detectable for the resulting von Neumann algebra.  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 group theoretic properties, such as direct products, can be recovered from the algebra.  We will also discuss recent progress made by Ben Hayes, Dan Hoff, Thomas Sinclair and myself in the case where the underlying group has positive first $\ell^2 $-Betti number.  We will explore the relationship between s-malleable deformations of von Neumann algebras and $\ell^2 $ co-cycles which lays the foundation for our work.