Free degrees of homeomorphisms on compact surfaces

Abstract

For a compact surface $M$, the free degree $\mathfrak{f}\mathfrak{r}\left(M\right)$ of homeomorphisms on $M$ is the minimum positive integer $n$ with property that for any self homeomorphism $\xi$ of $M$, at least one of the iterates $\xi ,{\xi }^2,\dots ,{\xi }^n$ has a fixed point. This is to say $\mathfrak{f}\mathfrak{r}\left(M\right)$ is the maximum of least periods among all periodic points of self homeomorphisms on $M$. We prove that $\mathfrak{f}\mathfrak{r}\left(F_{g,b}\right)\le 24g-24$ for $g\ge 2$ and $\mathfrak{f}\mathfrak{r}\left(N_{g,b}\right)\le 12g-24$ for $g\ge 3$.