For more information about this meeting, contact Robert Vaughan.

Title: | Arithmetic groups vs. mapping class groups: similarities and differences |

Seminar: | Algebra and Number Theory Seminar |

Speaker: | Lizhen Ji, University of Michigan |

Abstract: |

One of the most basic group actions is the action of the modular group SL(2, Z) on the upper half-plane H^2. The upper half plane H^2 can be canonically identified with the moduli space of marked elliptic curves (or rather compact Riemann surfaces of genus 1), and also with the moduli space of marked lattices in R^2 of covolume 1. The quotient SL(2, Z) \ H^2 is the moduli space of elliptic curves.
It admits two very important generalizations:
(1) SL(n, R)/SO(n), the space of marked lattices of covolume 1 in R^n, and the action of SL(n, Z) on it. More generally, actions of arithmetic groups on symmetric spaces of noncompact type.
(2) the Teichmuller space T_g of marked compact Riemann surfaces of genus g
at least 2, and the action of mapping class group Mod_g on T_g by changing the markings.
The quotient in case (1) is the space of equivalence class of positive definite
quadratic forms of determinant 1, and in case (2) is the moduli space M_g of projective curves of genus g. In this talk, I will discuss several results showing similarities and differences of these two generalizations. In particular, I will describe a generalization of the Minkowski reduction theory for lattices (or quadratic forms) in the theory of geometry of numbers to marked Riemann surfaces in T_g and use it to solve a long-standing folklore open problem in geometry and topology concerning the action of Mod_g on T_g. |

### Room Reservation Information

Room Number: | MB106 |

Date: | 04 / 09 / 2009 |

Time: | 11:15am - 12:05pm |