Tame Almost Primitive Elements

Abstract

ω is called a primitive element in a finitely generated group G if there is a minimal generating system of G which contains ω. An almost primitive element ω; in G is an element which is primitive in each finitely generated proper subgroup of G containing ω; and an almost primitive element ω ∈ G is called a tame almost primitive element whenever ωα is contained in a finitely generated subgroup H of G with α ≥ 1 minimal then either ωα is primitive in H or the index of H in G is just α. In this paper we consider for G the free group 〈a1, …, an;〉 of rank n ≥ 2 and the surface group 〈a1, …, an; [a1, a2]…[an−1, an] = 1〉, n ≥ 4 even. We show especially that the product a1p1 … anpn, pi ≥ 2, of powers of the generators is a tame almost primitive element if and only if p1 = … = pn = 2. 1.