Dehn Twists and Products of Mapping Classes of Riemann Surfaces with One Puncture |
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Authors: | Chaohui ZHANG |
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Institution: | Department of Mathematics, Morehouse College, Atlanta, GA 30314, USA |
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Abstract: | Let S be a Riemann surface that contains one puncture x. Let ℐ be the collection of simple closed geodesics on S, and let ℱ denote the set of mapping classes on S isotopic to the identity on S ∪ {x}. Denote by t
c
the positive Dehn twist about a curve c ∈ ℐ. In this paper, the author studies the products of forms (t
b
−m
∘ t
a
n
) ∘ f
k
, where a, b ∈ ℐ and f ∈ 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 ∈ ℱ such that the sequence (t
b
−m
∘ t
n
a
) ∘ f
k
contains infinitely many powers of Dehn twists. The author shows that the converse statement remains true, that is, if the
sequence (t
b
−m
∘ t
a
n
) ∘ 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 ∘ t
a
. |
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Keywords: | Riemann surfaces Simple closed geodesics Dehn twists Products Bers isomorphisms |
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