Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-05T15:24:37.819Z Has data issue: false hasContentIssue false
  • English
  • Français

ON PRODUCTS OF PSEUDO-ANOSOV MAPS AND DEHN TWISTS OF RIEMANN SURFACES WITH PUNCTURES

Part of: Deformations of analytic structures

Published online by Cambridge University Press:  26 April 2010

C. ZHANG
C. ZHANG*
Affiliation:
Department of Mathematics, Morehouse College, Atlanta, GA 30314, USA (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let S be a Riemann surface of type (p,n) with 3p+n>4 and n≥1. We investigate products of some pseudo-Anosov maps θ and Dehn twists tα on S, and prove that under certain conditions the products tkα∘θ are pseudo-Anosov for all integers k. We also give examples that show that tkα∘θ are not pseudo-Anosov for some integers k.

Keywords

Riemann surfacesDehn twistspseudo-Anosov mapsTeichmüller spaces

MSC classification

Secondary: 32G15: Moduli of Riemann surfaces, Teichmüller theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 88 , Issue 3 , June 2010 , pp. 413 - 428
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[1]Bauer, M., ‘Examples of pseudo-Anosov homeomorphisms’, Trans. Amer. Math. Soc. 330 (1992), 333–370.Google Scholar
[2]Bers, L., ‘Fiber spaces over Teichmüller spaces’, Acta Math. 130 (1973), 89–126.Google Scholar
[3]Boyer, S., Gordon, C. and Zhang, X., ‘Dehn fillings of large hyperbolic 3-manifolds’, J. Differential Geom. 58 (2001), 263–308.CrossRefGoogle Scholar
[4]Fathi, A., ‘Dehn twists and pseudo-Anosov diffeomorphisms’, Invent. Math. 87 (1987), 129–151.CrossRefGoogle Scholar
[5]Fathi, A., Laudenbach, F. and Poenaru, V., ‘Travaux de Thurston sur les surfaces’, in: Seminaire Orsay, Asterisque, Soc. Math. de France, 66–67 (1979), 1–284.Google Scholar
[6]Ivanov, N. V., ‘Mapping class groups’, Manuscript, 1998.Google Scholar
[7]Kra, I., ‘On the Nielsen–Thurston–Bers type of some self-maps of Riemann surfaces’, Acta Math. 146 (1981), 231–270.Google Scholar
[8]Long, D. D., ’Constructing pseudo-Anosov maps’, in: Knot Theory and Manifolds, Lecture Notes in Mathematics, 1144 (Springer, Berlin, 1985), pp. 108–114.Google Scholar
[9]Long, D. D. and Morton, H., ‘Hyperbolic 3-manifolds and surface homeomorphism’, Topology 25 (1986), 575–583.Google Scholar
[10]Masur, H. and Minsky, Y., ‘Geometry of the complex of curves I: Hyperbolicity’, Invent. Math. 138 (1999), 103–149.CrossRefGoogle Scholar
[11]Nag, S., ‘Non-geodesic discs embedded in Teichmüller spaces’, Amer. J. Math. 104 (1982), 339–408.CrossRefGoogle Scholar
[12]Penner, R. C., ‘The action of the mapping class group on isotopy classes of curves and arcs in surfaces’, Thesis, MIT, 1982.Google Scholar
[13]Penner, R. C., ‘A construction of pseudo-Anosov homeomorphisms’, Proc. Amer. Math. Soc. 104 (1988), 1–19.Google Scholar
[14]Thurston, W. P., ‘On the geometry and dynamics of diffeomorphisms of surfaces’, Bull. Amer. Math. Soc. (N.S.) 19 (1988), 417–431.CrossRefGoogle Scholar
[15]Zhang, C., ‘Pseudo-Anosov maps and fixed points of boundary homeomorphisms compatible with a Fuchsian group’, Osaka J. Math. 46 (2009), 783–798.Google Scholar
[16]Zhang, C., ‘Singularities of quadratic differentials and extremal Teichmüller mappings defined by Dehn twists’, J. Aust. Math. Soc. 3 (2009), 275–288.Google Scholar
[17]Zhang, C., ‘Pseudo-Anosov mapping classes and their representations by products of two Dehn twists’, Chinese Ann. Math. Ser. B 30 (2009), 281–292.CrossRefGoogle Scholar