# On algebraic extensions and decomposition of homomorphisms in free groups

## TL;DR

This paper provides a counterexample to a conjecture about algebraic extensions in free groups, showing that the relationship between algebraic extensions and core graph morphisms is more complex than previously thought.

## Contribution

The paper introduces a counterexample to a conjecture linking algebraic extensions of free groups with onto core graph morphisms, and presents a new partition of homomorphisms between free groups.

## Key findings

- Counterexample disproves the conjecture.
- Partition of homomorphisms offers new insights.
- Highlights complexity in free group extensions.

## Abstract

We give a counterexample to a conjecture by Miasnikov, Ventura and Weil, stating that an extension of free groups is algebraic if and only if the corresponding morphism of their core graphs is onto, for every basis of the ambient group. In the course of the proof we present a partition of the set of homomorphisms between free groups which is of independent interest.