Virtually free finite-normal-subgroup-free groups are strongly verbally closed

TL;DR

This paper proves that virtually free groups with no non-trivial finite normal subgroups are strongly verbally closed, meaning they are retracts of any finitely generated group containing them as verbally closed subgroups.

Contribution

It establishes a new property of virtually free groups, showing they are strongly verbally closed under certain conditions, expanding understanding of subgroup embeddings.

Findings

Virtually free groups with no non-trivial finite normal subgroups are strongly verbally closed.

Such groups are retracts of any finitely generated group containing them as verbally closed subgroups.

Includes examples like the infinite dihedral group.

Abstract

Any virtually free group $H$ containing no non-trivial finite normal subgroup (e.g., the infinite dihedral group) is a retract of any finitely generated group containing $H$ as a verbally closed subgroup.