# A remark on $\varepsilon$-maps in dimension 1

## TL;DR

This paper investigates whether certain surjective maps from the circle to a graph, with small pre-image diameters, can be decomposed as free factors in the fundamental group, exploring properties of epsilon-maps in one dimension.

## Contribution

It examines conditions under which epsilon-maps from the circle to a graph split as free factors in the fundamental group, providing insights into their algebraic and topological structure.

## Key findings

- Identifies conditions for epsilon-maps to split as free factors.
- Establishes bounds on epsilon for such splittings.
- Provides examples and counterexamples related to epsilon-maps.

## Abstract

Let $f\colon \mathbb{S}^1\rightarrow G$ be a surjective map from the standard unit circle to a graph $G$ such that the pre-image of each point has diameter less than $\varepsilon$. If $\varepsilon$ is small enough, does $f$ split as a free factor in $\pi_1(G)$?