Lifting closed curves to finite covers of free groups

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.