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 complex hyperbolic. A remarkable construction 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.