Nielsen generating sets and quasiconvexity of subgroups Original Research Article

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 set membership, variant H and for any shortest word w representing g there exist si set membership, variant S, 1 ≤ i ≤ m and a decomposition si ≡ liniri with ninot identical with1 such that g = s1 … sm and w ≡ l1n1n2 … nmrm. 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 ≥ 0 the set of K-quasiconvex subgroups of G is finite.