Counting groups, manifolds and primes

We will report on a series of works in the last 2  decades aroound   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. A recent joint work (joint with M. Belolipetsky) gives a sharp estimate for the number of non unoform lattices in high rank simple groups.