Mathematics – Geometric Topology
Scientific paper
2011-05-09
Mathematics
Geometric Topology
14 pages
Scientific paper
Let $S_n$ be a punctured Riemann spheres $\mathbf{S}^2\backslash \{x_1,..., x_n\}$. In this paper, we investigate pseudo-Anosov maps on $S_n$ that are isotopic to the identity on $S_n\cup \{x_n\}$ and have the smallest possible dilatations. We show that those maps cannot be obtained from Thurston's construction (that is the products of two Dehn twists). We also prove that those pseudo-Anosov maps $f$ on $S_n$ with the minimum dilatations can never define a trivial mapping class as any puncture $x_i$ of $S_n$ is filled in. The main tool is to give both lower and upper bounds estimations for dilatations $\lambda(f)$ of those pseudo-Anosov maps $f$ on $S_n$ isotopic to the identity as a puncture $x_i$ of $S_n$ is filled in.
Zhang Chaohui
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