On Rationality of Verbal Subsets In a Group

TL;DR

This paper characterizes when the set of all values of a group word in a free non-abelian group is rational, showing it occurs only in trivial cases or when it equals the entire group, and extends results to free products.

Contribution

It provides a complete characterization of rational verbal subsets in free groups and generalizes the result to free products of groups.

Findings

The set of all values of a group word in a free non-abelian group is rational only if it is trivial or the whole group.

The result is extended to a broad class of free products of groups.

The paper establishes a clear criterion linking rationality of verbal subsets to triviality or universality.

Abstract

Let $F$ be a free non-abelian group. We show that for any group word $w$ the set $w[F]$ of all values of $w$ in $F$ is rational in $F$ if and only if $w[F] = 1$ or $w[F] = F.$ We generalize this to a wide class of free products of groups.