Lifting closed curves to finite covers of free groups

TL;DR

This paper proves that any closed curve on a bouquet of circles can be lifted to a finite normal cover of any given degree, showing that free groups are unions of finite index normal subgroups.

Contribution

It provides an explicit construction of finite covers and characterizes when a closed curve lifts to these covers, establishing a new result about free groups and their subgroups.

Findings

Any closed curve on a bouquet of circles lifts to a finite l-sheeted normal cover.

Constructs explicit families of l-sheeted normal covers.

Characterizes conditions for lifting curves to these covers.

Abstract

In this article, we show that given any integer $l\geq 2$, every closed curve $\gamma$ on the bouquet of $n$-circles $\Gamma$, admits a lift to a finite $l$-sheeted normal covering of $\Gamma$. Equivalently, identifying the free group $F_n$ of $n$ generators with the fundamental group of $\Gamma$, this statement asserts that $F_n$ is a union of $ l$-index normal subgroups for any $l\geq 2.$ The proof proceeds by explicitly constructing families of $l$-sheeted normal coverings of $\Gamma$, together with a characterization, in terms of necessary and sufficient conditions, of when a closed curve $\gamma$ on $\Gamma$ lifts to these covers.