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Dergiler / Turkish Journal of Mathematics / 2010 / Cilt: 34 - Sayı: 4

Number of pseudo–Anosov elements in the mapping class group of a four–holed sphere

Ferihe ATALAN
Mustafa KORKMAZ
Sayfa
585–592
Yayın Tarihi
2022-11-03
DOI
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⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

We prove that the ratio of the number of pseudo-Anosov elements to that of all elements in a ball with center at the identity tends to one as the radius of the ball tends to infinity.

📌 3 Maddede Bu Makale

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Amaç & Hipotez:We compute the growth series and the growth functions of reducible and pseudo-Anosov elements of the pure mapping class group of the sphere with four holes with respect to a certain generating set.
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Temel Bulgular & Veri:We compute the growth series and the growth functions of reducible and pseudo-Anosov elements of the pure mapping class group of the sphere with four holes with respect to a certain generating set.
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Sonuç & Katkı:We prove that the ratio of the number of pseudo-Anosov elements to that of all elements in a ball with center at the identity tends to one as the radius of the ball tends to infinity.
🔬Tür:Araştırma Makalesi
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Alan Parametreleri

Abstract

We compute the growth series and the growth functions of reducible and pseudo-Anosov elements of the pure mapping class group of the sphere with four holes with respect to a certain generating set. We prove that the ratio of the number of pseudo-Anosov elements to that of all elements in a ball with center at the identity tends to one as the radius of the ball tends to infinity.

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Bu Makaleye Atıf Yap

ATALAN, F., KORKMAZ, M. (2022). Number of pseudo–Anosov elements in the mapping class group of a four–holed sphere. Turkish Journal of Mathematics, Cilt: 34 - Sayı: 4, 585–592. https://ajindex.com/articles/608514

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Kaynakça (7)

  1. Casson, A.J. and Bleiler, S.A.: Automorphisms of Surfaces After Nielsen and Thurston, Cambridge University Press 1988
  2. Farb, B.: Some problems on mapping class groups and moduli space in ”Problems on Mapping Class Groups and Related Topics”, ed. by B. Farb, Proc. Symp. Pure and Applied Math. 74, 11-55 2006
  3. Harpe, P.: Topics in Geometric Group Theory, The University of Chicago Press, Chicago 2000
  4. Ivanov, N.V.: Automorphisms of Teichmüller modular groups, Lecture Notes in Math., Springer, Berlin, 1346, 199-270 1988
  5. Maher, J.: Random walks on the mapping class group, arXiv.math/0604433
  6. Rivin, I.: Walks on groups, counting reducible matrices, polynomials, and surface and free group automorphims, Duke Math.J. 142(2), 353-379 (2008)
  7. Takasawa, M.: Enumeration of Mapping Classes for the Torus, Geometriae Dedicata 85, 11-19 (2001).