Quantifying Brinkmann's problem: relative $\varphi$-order and $\varphi$-spectrum

TL;DR

This paper investigates the computability of the relative $$-order and $$-spectrum in virtually free and finitely presented groups, providing algorithms for specific cases and extending to virtually abelian groups.

Contribution

It introduces the concept of relative $$-order and $$-spectrum, proving their computability in certain classes of groups and extending previous results.

Findings

The $$-spectrum is computable for finite subsets in virtually free groups.

The $$-spectrum is computable for recognizable subsets in finitely presented groups.

The paper discusses extensions to finitely generated virtually abelian groups.

Abstract

We prove that the stable image of an endomorphism of a virtually free group is computable. For an endomorphism $\varphi$, an element $x\in G$ and a subset $K\subseteq G$, we say that the relative $\varphi$-order of $g$ in $K$, $\varphi\text{-ord}_K(g)$, is the smallest nonnegative integer $k$ such that $g\varphi^k\in K$. We prove that the set of orders, which we call $\varphi$-spectrum, is computable in two extreme cases: when $K$ is a finite subset and when $K$ is a recognizable subset. The finite case is proved for virtually free groups and the recognizable case for finitely presented groups. The case of finitely generated virtually abelian groups and some variations of the problem are also discussed.