# Local to global property in free groups

## TL;DR

This paper investigates the local to global property in free groups, demonstrating that solvability in all finite quotients implies solvability in the free group itself, with specific focus on m-powers.

## Contribution

It establishes that for free groups, verifying the property for groups with fewer generators suffices to prove it generally, and applies this to m-powers.

## Key findings

- Proves local to global property for free groups with a reduction to fewer generators.
- Shows the property holds for m-powers in free groups.
- Provides a new approach to understanding equations over free groups.

## Abstract

The local to global property for an equation $\psi$ over a group G asks to show that $\psi$ is solvable in G if and only if it is solvable in every finite quotient of G. In this paper we focus that in order to prove this local to global property for free groups $G=F_k$, it is enough to prove for k less or equal the number of parameters in $\psi$. In particular we use it to show that the local to global property holds for m-powers in free groups.