Tokyo Journal of Mathematics

Crystallographic Groups Arising from Teichmüller Spaces

Yukio MATSUMOTO

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Abstract

The purpose of this paper is to show that there is a natural connection between Teichmüller spaces and crystallographic groups. This connection was first discovered by the author when discussing the Deligne-Mumford compactification of moduli spaces (see \lq\lq {\it The Deligne-Mumford compactification and crystallographic groups}\rq\rq, preprint, 2018). In this paper, we will discuss the same connection from a slightly different viewpoint having in mind an analogy of the maximally degenerate frontier points of the augmented Teichmüller spaces and the cusp points of hyperbolic spaces.

Article information

Source
Tokyo J. Math., Volume 42, Number 1 (2019), 1-34.

Dates
First available in Project Euclid: 18 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1563436911

Mathematical Reviews number (MathSciNet)
MR3982047

Zentralblatt MATH identifier
07114898

Subjects
Primary: 30F60: Teichmüller theory [See also 32G15]
Secondary: 20H15: Other geometric groups, including crystallographic groups [See also 51-XX, especially 51F15, and 82D25]

Citation

MATSUMOTO, Yukio. Crystallographic Groups Arising from Teichmüller Spaces. Tokyo J. Math. 42 (2019), no. 1, 1--34. https://projecteuclid.org/euclid.tjm/1563436911


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References

  • W. Abikoff, Augmented Teichmüller spaces, Bull. Amer. Math. Soc. 82 (1971), 333–334.
  • W. Abikoff, The real analytic theory of Teichmüller space, Lecture Notes in Math. 820, Springer-Verlag, 1976.
  • L. V. Ahlfors, Some remarks on Teichmüller's space of Riemann surfaces, Ann. Math. 74 (1961), 171–191.
  • L. V. Ahlfors, Curvature properties of Teichmüller space, J. d'Analyse Math. 9 (1961), 161–176.
  • L. V. Ahlfors, Lectures on Quasiconformal Mappings, D. Van Nostrand Company, Inc., Princeton, N. J. 1966. Wadsworth Inc., Belmont, California 94002, 1987.
  • L. V. Ahlfors and L. Bers, Riemann mapping theorem for variable metrices, Ann. Math. 72 (1960), 385–404.
  • L. Bers, Simultaneous uniformization, Bull. Amer. Math. Soc. 66 (1960), 94–97.
  • N. Bourbaki, Groupes et algèbres de Lie, Chapitres 4, 5 et 6, Éléments de Mathématique, Hermann, 1968.
  • H. Brown, R. Bülow, J. Neubüser, H. Wondratschek and H. Zassenhaus, Crystallographic groups of four-dimensional space, Wiley monographs in crystallography, John Wiley & Sons, Inc., 1978.
  • T. Chu, The Weil-Petersson Metric in the moduli space, Chinese J. Math. 4 (1976), 29–51.
  • C. J. Earle and I. Kra, On holomorphic mappings between Teichmüller spaces, in Contributions to Analysis (ed. by L. V. Ahlfors, et al.), Academic Press, New York, 1974, 107–124.
  • F. P. Gardiner, Schiffer's interior variation and quasiconformal mapping, Duke Math. J. 42 (1975), 371–380.
  • W. Harvey, Geometric structure of surface mapping class groups, in Homological Group Theory (ed. by C. T. C. Wall), London Math. Soc. Lecture Note Series 36 (1979), 255–269.
  • J. H. Hubbard, Teichmüller theory, Vol. 1, Matrix Editions, 2006.
  • Y. Imayoshi and M. Taniguchi, An introduction to Teichmüller spaces, Springer-Verlag, 1992.
  • A. Ishida, The structure of subgroup of mapping class groups generated by two Dehn twists, Proc. Japan Acad. 72 Ser. A (1996), 240–241.
  • B. Iversen, Lectures on crystallographic groups, Lecture Notes Series 1990/91 No. 60, Matematisk Institut, Aarhus Universitet.
  • T. Kawasaki, Monyou no kikagaku (Geometry of patterns), Makino Shoten, Tokyo, 2014.
  • L. Keen, Collars on Riemann surfaces, in Discontinuous Groups and Riemann surfaces, Proc. of the 1973 Conf. at Univ. of Maryland (ed. by L. Greenberg), Ann. Math. Studies 79 (1974), 263–268.
  • T. Kohno, Kesshô gun (Crystallographic groups), Kyoritsu Shuppan, Tokyo, 2015.
  • Y. Matsumoto, The Deligne-Mumford compactification and crystallographic groups, preprint, 2018.
  • Y. Matsumoto and J. M. Montesinos-Amilibia, A proof of Thurston's uniformization theorem of geometric orbifolds, Tokyo J. Math. 14 (1991), 181–196.
  • Y. Matsumoto and J. M. Montesinos-Amilibia, Pseudo-periodic homeomorphisms and degeneration of Riemann surfaces, Bull. Amer. Math. Soc. 30 (1994), 70–75.
  • Y. Matsumoto and J. M. Montesinos-Amilibia, Pseudo-periodic maps and degeneration of Riemann surfaces, Lecture Notes in Math. 2030, Springer, 2011.
  • H. Miyachi, Unification of extremal length geometry on Teichmüller space via intersection number, Math. Zeitschrift, 278 (2014), 1065–1095.
  • J. M. Montesinos-Amilibia, Grupos crystalográficos y topología en Escher, Rev. R. Acad. Cienc. Exact Fis. Nat. (Esp) 104 No. 1, (2010), 27–47.
  • S. Nag, The complex analytic theory of Teichmüller spaces, Canadian Math. Soc. Series of Monographs and Advanced Texts, A Willey-Interscience Publication, 1988.
  • J. G. Ratcliffe, Foundations of Hyperbolic Manifolds (second edition), Graduate Texts in Math. 149, Springer, 2006.
  • H. L. Royden, Automorphisms and isometries of Teichmüller space, in Advances in the theory of Riemann surfaces (ed. by L. V. Ahlfors et al.), Ann. of Math. Studies 66 (1971), 369–383.
  • R. L. E. Schwarzenberger, $N$-dimensional crystallography, Pitnam Advanced Publishing Program, 1980.
  • A. Szczepański, About the possibilities of application of relative hyperbolization of polyhedra, Math. Nachr. 172 (1995), 283–290.
  • A. Szczepański, Eta invariants for flat manifolds, Ann. Global Anal. Geom. 41 (2012), no. 2, 125–138.
  • S. Wolpert, Noncompleteness of the Weil-Petersson metric for Teichmüller space, Pacific J. Math. 61 (1975), 573–577.
  • S. Wolpert, The Fenchel-Nielsen deformation, Ann. Math. 115 (1982), 501–528.
  • S. Wolpert, On the symplectic geometry of deformations of a hyperbolic surfaces, Ann. Math. 117 (1983), 207–234.
  • S, Wolpert, On the Weil-Petersson geometry of the moduli space of curves, Amer. J. Math. 107 (1985), 969–997.
  • S. Wolpert, Geodesic length functions and the Nielsen problem, J. Diff. Geom. 25 (1987), 275–296.
  • S. Wolpert, Behavior of geodesic-length functions on Teichmüller space, J. Diff. Geom. 79 (2008), 277–334.
  • S. Wolpert, Families of Riemann surfaces and Weil-Petersson Geometry, Regional Conference Series in Mathematics 113, Amer. Math. Soc., 2010.
  • S. Yamada, On the geometry of Weil-Petersson completion of Teichmüller spaces, Math. Research Lett. 11 (2004), 327–344.