arXiv subjects
Publications and source records attributed to Nicolas Bergeron.
Let M be an arithmetic hyperbolic 3-manifold, such as a Bianchi manifold. We conjecture that there is a basis for the second homology of M, where each basis element is represented by a surface of `low' genus, and give evidence for this. We explain the relationship between this conjecture and the study of torsion homology growth.
We study the asymptotic growth of Betti numbers in tower of finite covers and provide simple proofs of approximation results, which were previously obtained by Calegari-Emerton, in the generality of arbitrary p-adic analytic towers of covers. Further, we also obtain partial results about arbitrary pro-$p$ towers.
This is the text of my Bourbaki seminar on the proof of the surface subgroup conjecture by Jeremy Kahn and Vladimir Markovic.
In the paper we define a "volume" for simplicial complexes of flag tetrahedra. This generalizes and unifies the classical volume of hyperbolic manifolds and the volume of CR tetrahedra complexes. We describe when this volume belongs to the Bloch group. In doing so, we recover and generalize results of Neumann-Zagier, Neumann, and Kabaya. Our approach is very related to the work of Fock and Goncharov.
Generalizing the proof -- by Hecht and Schmid -- of Osborne's conjecture we prove an Archimedean (and weaker) version of a theorem of Colette Moeglin. The result we obtain is a precise Archimedean version of the general principle -- stated by the second author -- according to which {\it a local Arthur packet contains the corresponding local $L$-packet and representations which are more tempered.}
We announce new results concerning the asymptotic behavior of the Betti numbers of higher rank locally symmetric spaces as their volumes tend to infinity. Our main theorem is a uniform version of the Lück Approximation Theorem \cite{luck}, which is much stronger than the linear upper bounds on Betti numbers given by Gromov in \cite{BGS}. The basic idea is to adapt the theory of local convergence, originally introduced for sequences of graphs of bounded degree by Benjamimi and Schramm, to sequences of Riemannian manifolds. Using rigidity theory we are able to show that when the volume tends to infinity, the manifolds locally converge to the universal cover in a sufficiently strong manner that allows us to derive the convergence of the normalized Betti numbers.
When does the amount of torsion in the homology of an arithmetic group grow exponentially with the covolume? We give many examples where this is so, and conjecture precise conditions.
En nous basant sur les résultats d'Arthur annoncés dans \cite[§30]{Arthur} nous démontrons les conjectures énoncées dans \cite{IMRN,BC,SMF} dans le cas des groupes orthogonaux à l'exclusion des groupes de type ${}^6 - D_4$. En ce qui concerne ces derniers, nous annonçons la démonstration -- encore en préparation - que leur réseaux de congruences ont toujours un $H^1$ trivial. Les démonstrations d'Arthur devraient para\^ıtre prochainement.
We give a criterion in terms of the boundary for the existence of a proper cocompact action of a word-hyperbolic group on a CAT(0) cube complex. We describe applications towards lattices and hyperbolic 3-manifold groups. In particular, by combining the theory of special cube complexes, the surface subgroup result of Kahn-Markovic, and Agol's criterion, we find that every subgroup separable closed hyperbolic 3-manifold is virtually fibered.