Abstract:

We will report on a series of works in the last decade around the following question:

"For a given simple Lie group G, how many lattices (i.e., discrete subgroups of finite covolume) does it have of covolume at most x?" Equivalently: "How many manifolds (of volume at most x) are covered by the associated symmetric space.?"

As many of these lattices are arithmetic, these questions often lead to deep number theoretic problems: counting primes etc.

We will concentrate on recent works which give very sharp results for counting arithmetic lattices in SL(2). Here the representation theory of the symmetric groups also comes into the game.