The proposed research includes two main projects. The first one, counting manifolds, is in a sense a continuous analog of asymptotic group theory. The guiding line is to convert finiteness results about locally symmetric manifolds, which follow from rigidity phenomenon and arithmeticity, to concrete quantitative statements. It is closely related to some central questions in mathematics such as the congruence subgroup problem, and to the remarkable finiteness theorem of Borel and Prasad, and it has applications in Riemannian geometry, number theory and theoretical physics. This project continues earlier work of the P.I. and joint work of the P.I. with Burger, Lubotzky and Mozes. The second main project concerns embeddings of free groups into groups with some geometric structure. This plays a central role in the study of linear and topological groups (in particular subgroups of Lie groups over local fields), and impacts some topics in differential geometry, ergodic theory, geometric group theory, unitary representations and profinite groups. One target, which the P.I. pursues in collaboration with E. Breuillard, is to obtain an effective version to Tits alternative, a weak version of which was proved by Eskin, Mozes and Oh, while solving Gromov's exponential growth conjecture. Other problems are related to the Auslander conjecture. This project is also related to the study of dense subgroups of analytic Lie groups, and the ``opposite'' problem of classifying the (analytic) metric completions of a given countable group.
There are several classical finiteness statement concerning locally symmetric spaces which have been known for more than 30 years, and yet have no quantitative proofs, or for which the existing estimates are suboptimal. One example is the classical theorem of Wang (and its strong version due to Borel and Prasad) about the finiteness of the number of manifolds with bounded volume; we would like to have good estimates for this number. Another example is the fact that the fundamental group of a manifold with finite volume is finitely presented; the size of a minimal presentation can be estimated in terms of the volume. More generally, we study relations between the volume of manifolds and their geometric structure. The second project deals with free subgroups. In his celebrated 1972 paper J. Tits proved that any finitely generated linear group which is not virtually solvable contains a non-commutative free subgroup. This result, known today as the Tits alternative, answered a conjecture of Bass and Serre and was an important step toward the understanding of linear groups. Any improvement in Tits' theorem has immediate corollaries in various different fields of mathematics. The P.I. and E. Breuillard had recently established a topological version of Tits theorem which answered several questions in dynamics, Riemannian foliations and profinite groups.
