Residual Finiteness of Surface Groups

Abstract

It is known [2] that free groups, and more generally fundamental groups of 2-manifolds [1], are residually finite. We give here an elementary proof of these facts. Theorem. Let F be a (possibly bounded) 2-manifold, then tti(F) is residually finite. Proof. We may assume Fis compact and orientable. Given 1 =^a677i(F) we must find a normal subgroup of finite index in tti(F) which does not contain a. Since the intersection of all subgroups (of any finitely generated group) of a fixed finite index is normal and also of finite index, it suffices to show that for a (general position) map /:0s1, *)-+(F, *) representing a there is a finite sheeted covering p:F-^F such that/does not lift to a mapf:(S\*)^(P,*).