It is often fruitful to study an infinite discrete group via its finite quotients. For this reason, conditions that guarantee many finite quotients can be useful. One such notion is residual finiteness.
A group is residually finite if for any non-identity element g there is a homomorphism onto a finite group, which doesn’t map g to e. I will mention how this relates to topology, present an argument why the surface groups are residually finite and I’ll show that in this case it is enough to consider homomorphisms onto alternating groups.
- Junior Topology and Group Theory Seminar