Character subgroups of F-groups

Abstract

where Yi=AiB;A j 'B j '. Here mi is a rational integer ~ 2, s ~ 0, t ~ 0, and s + t + g > 0. The consid e ration of F-groups ar ises naturally in the s tudy of di sco ntinuous groups, s ince it is known that exce pt for certain trivial exceptions eve ry Fuchsian group is an F-group when considered as an abstract group and that, conve rsely, e ve ry F-group has a faithful represe ntation as a Fuchsian group. If G is a group, an additive character on G is any homomorphi sm X of G into the additive group of th e complex numbers. That is, X(Xy) = X(X) + X(y), for X and Y in G. By a character subgroup of G we mean any subgroup H of G s uc h that H is the kernel of some nontrivial add iti ve character on G. Othe rwi se s tated, there exists a nontrivial c haracter X on G such that I H = {XEG I X(X)= O} . Henceforth G will denote an F-group, ~ the normal closure in G of E 1 , •• • , Es , and G' the commutator subgroup of G. The purpose of th is note is to prove the following theorem s. THEOREM 1: A subgroup H of G is a character subgroup ofG if and onLy ifH is normal in G, ILJG'~, and G/H has no elements of finite order. THEOREM 2: A subgroup H of G is contained in a character subgroup of G if and only if H . G' ~ is of infinite index in G. 2. The essence of the proof of these theorems is contained in the two lemmas of this sec tion. LEMMA 1: G/ ~G' is a free abelian group on finitely many generators. PROOF: A simple calculation shows that G/~G' is isomorphi c to (G/~)/(G/~)' . (Th is holds for any normal subgroup ~ of G.) Suppose t ~ 1. Then eliminating the ge nerator PI from the presentation has the concomitant effect of removing the relation (TIYi)E, ... EsP1 •• • PI = 1. Thus G/~ is free of rank 2g+ t 1 and we can conclude that (G/~)/(G/~)' i s free abelian of rank 2g+ t 1.