The fundamental group of a surface, and a theorem of Schreier

Abstract

Schreier proved in [8] tha t a finitely generated normal subgroup U # { 1 } of a free group F is of finite index. This result was extended by Karrass and Solitar in [3], to the case when U is not necessarily normal, but contains a non-trivial normal subgroup of F. In Topology, the free groups occur as fundamental groups of surfaces with boundary, and we here extend the result still further (Theorem 6.1) to the case when F is the (non-abelian) fundamental group of any connected surface, with or without boundary, except for a Klein bottle. We use topological methods, and also the elements of Morse theory, although the latter could be ehminated. A sketch of this theory is included, however, par t ly for its intuitive appeal, and part ly because the Morse theory picks out stable generators of the fundamental group, and therefore is helpful as a tool. Indeed, the author was able to use it quickly to prove Schreier's Theorem, and Theorem 3.3 below (that an open surface has free fundamental group), before knowing tha t proofs already existed in the literature. Our exposition is always from the point of view tha t it is the surface, ra ther than the group, which is the ult imate object o f study; and we have perhaps laboured points tha t might irk the pure group-theorist.