Algebraic and context-free subsets of subgroups

TL;DR

This paper investigates the relationship between algebraic and context-free subsets within groups, establishing conditions under which certain properties hold and providing counterexamples to previous conjectures.

Contribution

It characterizes when a Fatou property holds for context-free subsets in groups and presents a counterexample related to algebraic subsets.

Findings

Fatou property holds for context-free subsets iff the group is virtually free

Counterexample to Herbst's question on algebraic subsets

Links between subgroup structure and subset properties

Abstract

We study the relation between the structure of algebraic and context-free subsets of a group G and that of a finite index subgroup H. Using these results, we prove that a kind of Fatou property, previously studied by Berstel and Sakarovitch in the context of rational subsets and by Herbst in the context of algebraic subsets, holds for context-free subsets if and only if the group is virtually free. We also exhibit a counterexample to a question of Herbst concerning this property for algebraic subsets.