School of Mathematical Sciences
Monday, May 18, 2015
Schreiber 006, 12:15
The Hebrew University of Jerusalem
Most hyperbolic manifolds are non-arithmetic
Abstract: A celebrated
result of Margulis asserts that finite volume locally symmetric
manifolds of rank >1 are arithmetic. Corlette and Gromov-Schoen
extended this result to rank one spaces, with the two exceptions of
real and the complex hyperbolic.
A remarkable paper of Gromov and Piatetski-Shapiro establishes the
existence of a non-arithmetic (real) hyperbolic manifold of finite
volume, in any given dimension.
We prove that in fact almost all hyperbolic manifolds are
non-arithmetic, with respect to a certain way of counting. Recall that
two manifolds are commensurable if they share a common finite cover.
Fixing the dimension d>3, and counting up to commensurability, we
show that the number of non-arithmetic hyperbolic d-manifolds of volume
bounded by V is super-exponential in V, while the number of arithmetic
ones tends to be polynomial.
This is a joint work with Arie Levit.
Coffee will be served at 12:00 before the lecture
at Schreiber building 006