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The Geometric Theory of the Fundamental Germ

Published online by Cambridge University Press:  11 January 2016

T. M. Gendron
T. M. Gendron*
Affiliation:
Instituto de Matemáticas - Unidad Cuernavaca, Universidad Nacional Autónoma de México, Av. Universidad s/n, C.P. 62210 Cuernavaca, Morelos, MÉXICO, [email protected]
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Abstract

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The fundamental germ is a generalization of π1, first defined for laminations which arise through group actions [4]. In this paper, the fundamental germ is extended to any lamination having a dense leaf admitting a smooth structure. In addition, an amplification of the fundamental germ called the mother germ is constructed, which is, unlike the fundamental germ, a topological invariant. The fundamental germs of the antenna lamination and the PSL(2,ℤ) lamination are calculated, laminations for which the definition in [4] was not available. The mother germ is used to give a new proof of a Nielsen theorem for the algebraic universal cover of a closed surface of hyperbolic type.

Keywords

57R30
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 190 , 2008 , pp. 1 - 34
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2008

References

[1] Bernik, V. I. and Dodson, M. M., Metric Diophantine Approximation on Manifolds, Cambridge Tracts in Mathematics 137, Cambridge University Press, Cambridge, 1999.Google Scholar
[2] Biswas, I., Nag, S. and Sullivan, D. P., Determinant bundles, Quillen metrics and Mumford isomorphisms over the universal commensurability Teichmüller space, Acta Math., 176 (1996), no. 2, 145–169.Google Scholar
[3] Gendron, T. M., Fuchsian Germs, PhD Thesis, City University of New York, 1997.Google Scholar
[4] Gendron, T. M., The algebraic theory of the fundamental germ, Bull. Braz. Math. Soc., (N.S.), 37 (2006), no. 1, 49–87.Google Scholar
[5] Gendron, T. M., The Ehrenpreis conjecture and the moduli-rigidity gap, Complex Manifolds and Hyperbolic Geometry (Earle et al, eds.), Contemporary Mathematics, AMS, Providence, RI, 2002, pp. 207–229.Google Scholar
[6] Ghys, E., Laminations par surfaces de Riemann, Dynamique et géométrie complexes (Lyon, 1997), Panoramas & Synthèse 8, Soc. Math. France, Paris, 1999, pp. 49–95.Google Scholar
[7] Goldbladt, R., Lectures on the Hyperreals, Springer-Verlag, New York, NY, 1998.Google Scholar
[8] Odden, C., The baseleaf preserving mapping class group of the universal hyperbolic solenoid, Trans. Amer. Math. Soc., 357 (2005), no. 5, 1829–1858.Google Scholar
[9] Robinson, A., Non-standard Analysis, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1996.Google Scholar
[10] Stroyan, K. D. and Luxemburg, W. A. J., Introduction to the theory of infinitesimals, Pure and Applied Mathematics, No. 72, Academic Press, New York-London, 1976.Google Scholar