Generating the liftable outer automorphism groups of regular cyclic covers

Abstract

Let $R_n = \bigvee_{i=1}^{n} S^1$ be a finite connected one-dimensional CW complex with $n \geq 2$, and let $\mathrm{Out}(F_n)$ be the outer automorphism group of the free group $F_n$, where $π_1(R_n)\cong F_n$. Let $p_k:\tilde{R}_n\rightarrow R_n$ be the standard $k$-sheeted regular cyclic cover of $R_n$ induced by a $\mathbb{Z}_k$-action on $\tilde{R}_n$ (which is homotopy equivalent to $R_{n_k}$, where $n_k=k(n-1)+1$) generated by a free rotation through $2π/k$ of $\tilde{R}_n$. Let $\mathrm{Out}_{F_{n_k}}(F_n)$ be the liftable outer automorphism group associated with a standard $k$-sheeted normal cover $p_k$. For $n\geq3$, we show that $\{\mathrm{Out}_{F_{n_k}}(F_n)\}_{k\geq2}$ forms an infinite family of self-normalizing subgroups of $\mathrm{Out}(F_n)$. For $n\geq2$, these groups are maximal if and only if $k$ is prime. Furthermore, we derive explicit finite generating sets for $\mathrm{Out}_{F_{n_k}}(F_n)$ for $k\geq 2$ and $n\geq 3$. Let $\mathcal{C}$ be the infinite chain of roses, and let $q_n:\mathcal{C}\longrightarrow R_n$ be the standard infinite-sheeted cover induced by $\langle τ\rangle$, where $τ$ is the unit translation on $\mathcal{C}$. As a final application, we obtain a finite generating set for $\mathrm{Out}_{F_\infty}(F_n)$ when $n\geq 3$.