Approximations of infinite groups

We discuss various (still open) questions on approximations of finitely generated groups, focusing on finite-dimensional approximations such as residual finiteness and soficity. We survey our results on the existence, stability and quantification of metric approximations. We suggest a few conjectures, e.g. on Gromov hyperbolic groups and their infinite monster limits.

Based on joint works with Liviu Paunescu (Bucharest).