# Two-generated verbally closed subgroups of a free solvable group G are retracts of G

## TL;DR

This paper proves that in free solvable groups of finite rank, any two-generated verbally closed subgroup is necessarily a retract of the entire group, revealing a structural property of these subgroups.

## Contribution

It establishes that all two-generated verbally closed subgroups in free solvable groups are retracts, a new insight into subgroup structure.

## Key findings

- Every two-generated verbally closed subgroup is a retract.
- The result applies to free solvable groups of finite rank.
- Provides a structural characterization of such subgroups.

## Abstract

We prove that every verbally closed two-generated subgroup of a free solvable group G of a finite rank is a retract of G.