Braids and Nielsen-Thurston Types of Automorphisms of Punctured Surfaces

Abstract

Let $F$ be a compact, orientable surface with negative Euler characteristic, and let $x_1, \cdots, x_n$ be $n$ fixed but arbitrarily chosen points on $\mathrm{int}F$, each of which has a (small) diskal neighborhood $D_i \subset F$. Denote by $\mathcal{S}_n(F)$ a subgroup of $\mathrm{Diff}(F)$ consisting of "sliding" maps $f$ each of which satisfies

(1) $f(\{x_1, \dots , x_n\}) = \{x_1, \dots , x_n\}, f(D_1\cup \cdots \cup D_n) = D_1 \cup \cdots \cup D_n $ and

(2) $f$ is isotopic to the identity map on $F$

Then by restricting such automorphisms to $\hat{F} = F - \mathrm{int}(D_1 \cup \cdots \cup D_n)$, we have automorphisms $\hat{f} : \hat{F} \to \hat{F}$, which form a subgroup $\mathcal{S}_n(\hat{F})$ of $\mathrm{Diff}(\hat{F})$. We give a Nielsen-Thurston classification of elements of $\mathcal{S}_n(\hat{F})$ using braids in $F \times I$ which characterize the elements of $\mathcal{S}_n(\hat{F})$.