Stability and Testability

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 $G$ is said to be flexibly stable if every sofic approximation to $G$ can converted to a sequence of disjoint unions of Schreier graphs on coset spaces of $G$ by modifying an asymptotically vanishing proportion of edges. We will discuss a joint result with Lewis Bowen that if $\mathrm{PSL}_d(\mathbb{Z})$ is flexibly stable for some $d \geq 5$ then there exists a group which is not sofic.