Flexible stability and nonsoficity

A sofic approximation to a countable discrete group is a sequence of finite models for the group that generalizes the concept of a Folner sequence witnessing amenability of a group and the concept of a sequence of quotients witnessing residual finiteness of a group. If a group admits a sofic approximation it is called sofic.

It is a well known open problem to determine if every group is sofic. A sofic group GG is said to be flexibly stable if every sofic approximation to GG can converted to a sequence of disjoint unions of Schreier graphs on coset spaces of GG by modifying an asymptotically vanishing proportion of edges. We will discuss a joint result with Lewis Bowen that if PSLd(ℤ)PSLd(Z) is flexibly stable for some d≥5d≥5 then there exists a group which is not sofic.