Tokyo Journal of Mathematics

Braids and Nielsen-Thurston Types of Automorphisms of Punctured Surfaces

Kazuhiro ICHIHARA and Kimihiko MOTEGI

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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 $\mathcal{S}_n(\hat{F})$.

Article information

Tokyo J. Math., Volume 28, Number 2 (2005), 527-538.

First available in Project Euclid: 5 June 2009

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces
Secondary: 57M50: Geometric structures on low-dimensional manifolds 57N10: Topology of general 3-manifolds [See also 57Mxx]


ICHIHARA, Kazuhiro; MOTEGI, Kimihiko. Braids and Nielsen-Thurston Types of Automorphisms of Punctured Surfaces. Tokyo J. Math. 28 (2005), no. 2, 527--538. doi:10.3836/tjm/1244208205.

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