Simple curves on surfaces and an analog of a theorem of Magnus for surface groups.
In 1931 W. Magnus proved that if two elements \(r_1\) and \(r_2\) in a free group \(F\) have the same normal closure in \(F\) then \(r_1\) is conjugate to \(r_2\) or \(r_2^{-1}\). Related to the Magnus theorem the authors proposed the following two questions: Problem 1. Does a (one-relator) group \(G\) have the following property: if \(u,v\) are elements of \(G\) with the same normal closure then \(u\) is a conjugate of \(v^{\pm 1}\)? Problem 2. Does an element \(u\) of a (one-relator) group \(G\) have the following property: if \(\varphi\) is an automorphism of \(G\) sending the normal closure of \(u\) into itself then \(\varphi(u)\) is conjugate of \(u^{\pm 1}\)? For those two questions the authors prove some interesting result in the case that \(G\) is the fundamental group of a closed surface \(S\) and \(g,h\) are nontrivial elements of \(\pi_1(S)\), both containing simple closed two-sided curves \(c_1\) and \(c_2\), respectively, i.e., if \(h\) belongs to the normal closure of \(g\), then \(h\) is conjugate to \(g^\varepsilon\) or to \((gug^\eta u^{-1})^\varepsilon\), \(\varepsilon,\eta\in\{-1,1\}\), where \(u\) is a homotopy class containing a simple closed curve \(\mu\) which properly intersects \(c_1\) exactly once. In particular, they show that if \(S\) is a closed surface different from the torus and the Klein bottle then every normal automorphism of \(\pi_1(S)\) is an inner automorphism.
scientific article; zbMATH DE number 2133496