Some results on one-relator surface groups

If S is noncompact, or has nonempty boundary, then π1(S) is free, and the answer to Question 1 is yes, by an old result of Magnus [7] on one-relator groups. (Essentially, the defining relator in a one-relator group on a given generating set is unique up to conjugacy and inversion.) We will show (see Theorem 3.4 below) that Question 1 also has an affirmative answer in the case of a closed surface S. In this case Question 1 can be interpreted in terms of one-relator surface groups, as introduced by Hempel [3]. Among other results, Hempel proved analogues for one-relator surface groups of two theorems from one-relator group theory: (i) a one-relator surface group is locally indicable if and only if the relator is not a proper power in π1(S); (ii) a closed curve α in S lifts (up to homotopy) to a simple closed curve in the covering space corresponding to the normal closure of α in π1(S). These are analogues of results of Brodskĭı[1] and Weinbaum [15] respectively. (In the latter case, the original form states that proper subwords of the defining relator represent nontrivial elements in a one-relator group.) Hempel [3] also proved (iii) that a power β of a simple closed curve β can belong to the normal closure in π1(S) of a curve α only in the obvious cases: either α is isotopic in S to β m with m|n; or n = 1 and α is a nonseparating curve in a punctured torus in S bounded by β.