# Verbally closed virtually free subgroups

## TL;DR

This paper generalizes a known result about verbally closed subgroups in free groups, showing they are retracts in broader classes of finitely generated groups, including virtually free groups.

## Contribution

It extends the characterization of verbally closed subgroups as retracts from free groups to all finitely generated groups, including many virtually free groups.

## Key findings

- Verbal closedness implies being a retract in finitely generated groups.
- All free and many virtually free verbally closed subgroups are retracts.
- Generalization of Myasnikov and Roman'kov's theorem to broader group classes.

## Abstract

A theorem of Myasnikov and Roman'kov says that any verbally closed subgroup of a finitely generated free group is a retract. We prove that all free (and many virtually free) verbally closed subgroups are retracts in any finitely generated group.