Counting Manifolds and Embeddings of Free Groups

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The proposed research includes two main projects. The first one, countingmanifolds, is in a sense a continuous analog of asymptotic group theory. The guiding line is to convert finiteness results about locally symmetricmanifolds, which follow from rigidity phenomenon and arithmeticity, toconcrete quantitative statements. It is closely related to some centralquestions in mathematics such as the congruence subgroup problem, and tothe remarkable finiteness theorem of Borel and Prasad, and it hasapplications in Riemannian geometry, number theory and theoreticalphysics. This project continues earlier work of the P.I. and joint work ofthe P.I. with Burger, Lubotzky and Mozes. The second main project concernsembeddings of free groups into groups with some geometric structure. Thisplays a central role in the study of linear and topological groups (inparticular subgroups of Lie groups over local fields), and impacts sometopics 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 effectiveversion to Tits alternative, a weak version of which was proved by Eskin,Mozes and Oh, while solving Gromov's exponential growth conjecture. Otherproblems are related to the Auslander conjecture. This project is alsorelated to the study of dense subgroups of analytic Lie groups, and the``opposite'' problem of classifying the (analytic) metric completions of agiven countable group.There are several classical finiteness statement concerning locallysymmetric spaces which have been known for more than 30 years, and yethave no quantitative proofs, or for which the existing estimates aresuboptimal. One example is the classical theorem of Wang (and its strongversion due to Borel and Prasad) about the finiteness of the number ofmanifolds with bounded volume; we would like to have good estimates forthis number. Another example is the fact that the fundamental group of amanifold with finite volume is finitely presented; the size of a minimalpresentation can be estimated in terms of the volume. More generally, westudy relations between the volume of manifolds and their geometricstructure. The second project deals with free subgroups. In his celebrated1972 paper J. Tits proved that any finitely generated linear group whichis not virtually solvable contains a non-commutative free subgroup. Thisresult, known today as the Tits alternative, answered a conjecture of Bassand Serre and was an important step toward the understanding of lineargroups. Any improvement in Tits' theorem has immediate corollaries invarious different fields of mathematics. The P.I. and E. Breuillard hadrecently established a topological version of Tits theorem which answeredseveral questions in dynamics, Riemannian foliations and profinite groups.