NIELSEN GENERATING SETS AND

QUASICONVEXITY OF SUBGROUPS

Rita Gitik

Abstract. Let H be a subgroup of G =< X|R >. We say that a finite generating set S of H is weakly Nielsen if for any g ∈ H and for any shortest word w representing g there exist s_{i} ∈ S, 1 ≤ i ≤ m and a decomposition s_{i} ≡ l_{i}n_{i}r_{i} with n_{i} ≢ 1 such that w ≡ l_{1}n_{1}n_{2} ⋯ n_{m}r_{m}.

We prove that a subgroup of a finitely generated group is quasiconvex if and only if it has a finite weakly Nielsen generating set, which implies that if a subgroup of a negatively curved group has a weakly Nielsen generating set, then it is negatively curved. We also prove that for any finitely generated group G and for any K ≥ 0 the set of K-quasiconvex subgroups of G is finite.

Typeset by AMS-TEX