Asymptotique de la norme $L^2$ d'un cycle géodésique dans les revêtements de congruence d'une variété hyperbolique arithmétique

Abstract

Let M be an arithmetic hyperbolic manifold and \(F\subset M\) be a codimension 1 geodesic cycle. In this paper, we study the asymptotic growth of the \(L^2\)-norm of the lifts of F in the congruence tower above M. We obtain an explicit value for the growth rate of this norm. In particular, we provide a new proof of a celebrated result of Millson [Mi] on the homology of the arithmetic hyperbolic manifolds. The method is quite general and gives a new way of getting non zero homology classes in certain locally symmetric spaces.