Simple curves on surfaces and an analog of a theorem of Magnus for surface groups

Abstract

Magnus proved that, if is a free group and u, v are elements of with the same normal closure, u is a conjugate of v or v −1 [9]. We prove the analogous result in the case that is the fundamental group of a closed surface S and u,v are elements of π1(S) containing simple closed two-sided curves on S. As a corollary we prove that, if S is not a torus and is not a Klein bottle, each automorphism of π1(S) which maps every normal subgroup of π1(S) into itself is an inner automorphism.