Is decidability of the Submonoid Membership Problem closed under finite extensions?

TL;DR

This paper investigates the decidability of the submonoid membership problem, showing reductions from rational subset membership in certain groups and providing evidence of potential non-closure under finite extensions.

Contribution

It introduces new reductions between rational subset and submonoid membership problems and explores the limits of decidability under finite index extensions.

Findings

No uniform reduction from submonoid to finite index subgroups in virtually nilpotent groups

Evidence suggesting possible non-closure of decidability under finite extensions

Reductions from rational subset membership to submonoid membership in specific group classes

Abstract

We show that the rational subset membership problem in $G$ can be reduced to the submonoid membership problem in $G{\times}H$ where $H$ is virtually Abelian. We use this to show that there is no algorithm reducing submonoid membership to a finite index subgroup uniformly for all virtually nilpotent groups. We also provide evidence towards the existence of a group $G$ with a subgroup $H<G$ of index 2, such that the submonoid membership problem is decidable in $H$ but not in $G$.