Relative order and spectrum in free and related groups

TL;DR

This paper explores a generalized notion of element order in groups, focusing on free groups, revealing both computability results and examples of complex spectra, including non-recursive sets, with implications for algorithmic group theory.

Contribution

It introduces and analyzes the concept of relative order in free groups, establishing its computability in certain cases and demonstrating the existence of complex, non-recursive spectra.

Findings

Order and spectrum are computable in free groups.

Existence of groups with arbitrary subsets of natural numbers as spectra.

Spectrum membership problem is unsolvable in certain direct products.

Abstract

In this paper, we consider a natural generalization of the concept of order of an element in a group: an element $g \in G$ is said to have order $k$ in a subgroup $H$ of $G$ (\resp \wrt a coset $Hu$) if $k$ is the first strictly positive integer such that $g^k \in H$ (\resp $g^k \in Hu$). We study this notion and its algorithmic properties in the realm of free groups and some related families. Both positive and negative (algorithmic) results emerge in this setting. On the positive side, among other results, we prove that the order of elements, the set of orders (called spectrum), and the set of preorders (\ie the set of elements of a given order) \wrt finitely generated subgroups are always computable in free and free times free-abelian groups. On the negative side, we provide examples of groups and subgroups having essentially any subset of natural numbers as relative spectrum; in particular, non-recursive and even non-recursively enumerable sets of natural numbers. Also, we take advantage of Mikhailova's construction to see that the spectrum membership problem is unsolvable for direct products of nonabelian free groups.