Abstract: Soficity is an intriguing approximation property for groups that was introduced in a seminal work of M. Gromov and singled out by B. Weiss, motivated by the
Gottschalk surjectivity conjecture in symbolic dynamics. Like other approximation properties preceding it, such as residual finiteness and amenability, soficity asks for the group to be approximated by finite groups, in a suitable sense. A weaker notion, called hyperlinearity, can be defined using approximation by unitary matrix groups. A big open question is to find examples (if they exist!) of non-sofic, or non-hyperlinear groups.
Recently, several new rigidity properties of groups, colloquially referred to as stability, have gathered considerable attention. Among them is flexible Hilbert Schmidt stability: A group G is flexibly HS-stable if any approximate finite dimensional unitary representation of G is close to a compression of a genuine representation of slightly larger dimension. In this talk, we will give conditional statements of the form "If G is flexibly HS-stable, then there exists a non hyperlinear group". This statement is shown to hold for various groups with property (T), including the integer symplectic group and Gromov random groups with probability 1. The proof technique is von Neumann algebraic in nature.