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| Dehn Twists and Products of Mapping Classes of Riemann Surfaces with One Puncture |
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Citation： |
Chaohui ZHANG.Dehn Twists and Products of Mapping Classes of Riemann Surfaces with One Puncture[J].Chinese Annals of Mathematics B,2011,32(6):885~894 |
| Page view： 1716
Net amount： 1213 |
Authors： |
Chaohui ZHANG; |
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| Abstract： |
Let $S$ be a Riemann surface that contains one puncture $x$. Let
$\mathscr{S}$ be the collection of simple closed geodesics on $S$,
and let $\mathscr{F}$ denote the set of mapping classes on $S$
isotopic to the identity on $S\cup \{x\}$. Denote by $t_c$ the
positive Dehn twist about a curve $c\in \mathscr{S}$. In this paper,
the author studies the products of forms $\left(t_b^{-m}\circ
t_a^n\right)\circ f^k$, where $a,b\in \mathscr{S}$ and $f\in
\mathscr{F}$. It is easy to see that if $a=b$ or $a,b$ are boundary
components of an $x$-punctured cylinder on $S$, then one may find an
element $f\in \mathscr{F}$ such that the sequence
$\left(t_b^{-m}\circ t_a^n\right)\circ f^k$ contains infinitely many
powers of Dehn twists. The author shows that the converse statement
remains true, that is, if the sequence $\left(t_b^{-m}\circ
t_a^n\right)\circ f^k$ contains infinitely many powers of Dehn
twists, then $a,b$ must be the boundary components of an
$x$-punctured cylinder on $S$ and $f$ is a power of the spin map
$t_b^{-1}\circ t_a$. |
Keywords： |
Riemann surfaces, Simple closed geodesics, Dehn twists, Products,
Bers isomorphisms |
Classification： |
32G15, 30C60, 30F60 |
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