Algebraic Geometry Seminar

Arithmetic Quotients of the Mapping Class Group

Speaker: Alex Lubotzky, Hebrew University of Jerusalem

Location: Warren Weaver Hall 201

Date: Tuesday, December 2, 2014, 3:30 p.m.


Let M=M(g) be the mapping class group of a surface of genus g>1. As it is well known, M is mapped onto the symplectic group Sp(2g,Z). We will show that this is only the first case in a series: in fact, to every pair (S,r), where S is a finite group with less than g generators and r is a Q-irreducible representation of S, we associate an arithmetic group which is then shown to be a virtual quotient of M. The case when S is the trivial group gives the above Sp(2g,Z), but many new quotients are obtained. For example, we show that M(2) is virtually mapped onto a non-abelian free group. Another application is an answer to a question of Kowalski: generic elements in the Torelli groups are hyperbolic and fully irreducible. Joint work with Fritz Gruenwald, Michael Larsen and Justin Malestein (to appear in GAFA).