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

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 ₁(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 ₁(S) which maps every normal subgroup of ₁(S) into itself is an inner automorphism.Mathematical Subject Classification (2000): 50F34, 57M07, 57M99The first two authors were supported by the DFG-Projekt Niedrigdimensionale Topologie und geometrisch-topologische Methoden in der Gruppentheorie. The first author was supported by the RFFI grant 02-01-99252 and the grant N7 of RAS in the 6-th competition of projects of young scientists. The second author was partially supported by the Support of Leading Scientifical Schools, 00-15-96059 (2000–2002) and by the RFFI grant 01-01-00583 (2001–2003). This work was partially done during the visits of the first and the second authors to the Fakultät für Mathematik, Ruhr-Universität Bochum, Germany in August 2001, and the visit of the third author to the mathematical department of the Moscow State University in 2001/02. This visit was supported by the Johann Gottfried Herder Stiftung.in final form: 18 July 2003