\'Equidistribution, comptage et approximation par irrationnels quadratiques

Abstract

Let $M$ be a finite volume hyperbolic manifold, we show the equidistribution in $M$ of the equidistant hypersurfaces to a finite volume totally geodesic submanifold $C$. We prove a precise asymptotic on the number of geodesic arcs of lengths at most $t$, that are perpendicular to $C$ and to the boundary of a cuspidal neighbourhood of $M$. We deduce from it counting results of quadratic irrationals over $\QQ$ or over imaginary quadratic extensions of $\QQ$, in given orbits of congruence subgroups of the modular groups.