Abstract:

"Subgroup growth" deals with counting finite index subgroups of a group. This theory led to counting congruence subgroups in arithmetic groups. The latter counting is a kind of "non-commutative analytic number theory" where "counting primes" on one hand and delicate finite group theory, on the other hand, are combined.

We will present the main counting results, applications to group theory and a connection with the congruence subgroup problem and the structure of the fundamental groups of hyperbolic manifolds.