Relative order and spectrum of subgroups

We consider a natural generalization of the concept of order of a (torsion) element: the order of $g\in G$ relative to a subgroup $H\leq G$ is the minimal $k>0$ such that $g^k\in H$; and the spectrum of $H$ is defined as the set of orders of elements from $G$ relative to $H$. After analyzing the first general properties of these concepts, we obtain the following results: (1) every set of natural numbers closed under divisors, is realizable as the spectrum of a finitely generated subgroup $H$ of a finitely generated torsion-free group $G$; (2) $F_n\times F_n$ has undecidable spectrum membership problem: there is no algorithm to decide, given a finitely generated subgroup $H$ and a natural number $k$, whether $k$ belongs to the spectrum of $H$; and (3): in free groups $F_n$ (as well as in free-times-free-abelian groups $F_n\times Z^m$) spectrum membership is solvable, and one can give an explicit algorithmic-friendly description of the set of elements of a given order $k$ relative to a given finitely generated subgroup $H$. (joint work with J. Delgado and A. Zakharov)