The decorated Teichmüller space of punctured surfaces
- 228 Downloads
A principal ℝ + 5 -bundle over the usual Teichmüller space of ans times punctured surface is introduced. The bundle is mapping class group equivariant and admits an invariant foliation. Several coordinatizations of the total space of the bundle are developed. There is furthermore a natural cell-decomposition of the bundle. Finally, we compute the coordinate action of the mapping class group on the total space; the total space is found to have a rich (equivariant) geometric structure. We sketch some connections with arithmetic groups, diophantine approximations, and certain problems in plane euclidean geometry. Furthermore, these investigations lead to an explicit scheme of integration over the moduli spaces, and to the construction of a “universal Teichmüller space,” which we hope will provide a formalism for understanding some connections between the Teichmüller theory, the KP hierarchy and the Virasoro algebra. These latter applications are pursued elsewhere.
KeywordsModulus Space Geometric Structure Quantum Computing Class Group Mapping Class
Unable to display preview. Download preview PDF.
- [Ab]Abikoff, W.: The real-analytic theory of Teichmüller space. Lecture Notes in Mathematics, Vol. 820. Berlin, Heidelberg, New York: Springer 1980Google Scholar
- [BE]Bowditch, B., Epstein, D.B.A.: Triangulations associated with punctured surfaces. Topology (1987)Google Scholar
- [Ca]Cassels, J.W.S.: Rational quadratic forms. New York: Academic Press 1978Google Scholar
- [EP]Epstein, D.B.A., Penner, R.C.: Euclidean decompositions of non-compact hyperbolic manifolds. J. Diff. Geom. (1987)Google Scholar
- [FS]Friedan, D., Shenker, S.H.: The analytic geometry of two dimensional conformal field theory. Nucl. Phys. B. (1987)Google Scholar
- [Ha]Harer, J.: The virtual cohomological dimension of the mapping class group of an oriented surface. Invent. Math84, 157–176 (1986)Google Scholar
- [HZ]Harer, J., Zagier, D.: The Euler characteristic of the moduli space of curves. Invent. Math.85, 457–485 (1986)Google Scholar
- [Mo]Mosher, L.: Pseudo-anosovs on punctured surfaces. Princeton University thesis (1983)Google Scholar
- [P1]Penner, R.C.: Perturbative series and the moduli space of Riemann surfaces. J. Diff. Geom. (1987)Google Scholar
- [P2]Penner, R.C.: The moduli space of punctured surfaces. Proceedings of the Mathematical Aspects of String Theory Conference, University of California, San Diego, World Science Press, 1987Google Scholar