Abstract: 
A group G is stable in permutations if every almost-action of G on a finite set is close to some actual action. Part of the interest in this notion comes from the observation that a non-residually finite stable group cannot be sofic. 
I will show that surface groups are stable in a flexible sense, that is if one is allowed to "add a few extra points" to the action. This is the first non-trivial stability result for a non-amenable group. 
The proof is essentially geometric. Along the way, we establish a quantitative variant of the LERF property for surface groups which may be of independent interest.
The talk is based on a joint work with Nir Lazarovich and Yair Minsky.