The set of finite quotients of a group can provide a lot of information about the group if this set is sufficiently rich. This is the case for a residually finite group, and studying its finite quotients geometrically has many implications for algebraic and analytic aspects of the group. One can also use this framework to create interesting examples of metric spaces. Rigidity results for finite quotients allow us to better understand the variety of examples obtained in this way. This talk will be based on joint work with Alain Valette, and with Thiebout Delabie.