Nielsen generating sets and quasiconvexity of subgroups

Let H be a subgroup of a group G. We say that a finite generating set S of H is weakly Nielsen if for any g is an element of H and for any shortest word w representing g there exist s(i) is an element of S, 1 less than or equal to i less than or equal to m and a decomposition s(i) = l(i)n(i)r(i) with n(i) not equivalent to 1 such that g = s(1)...s(m) and 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. It follows that the generalised word problem is solvable for locally quasiconvex negatively curved groups. We also prove that for any finitely generated group G and for any K greater than or equal to 0 the set of K-quasiconvex subgroups of G is finite.