Cycles géodésiques transverses dans les variétés hyperboliques

Abstract

Let M be a hyperbolic manifold, with \( \pi_1M \) finitely generated. Let c1 and c2 be two transverse geodesic cycles with dim(c1) + dim(c2) = dim M and \( c_1 \cap c_2 \neq \emptyset \). In this paper, following ideas of [MR], we prove (Theorem 6) that c1 and c2 lifts to a finite cover of M as two submanifolds F1 and F2 with¶¶\( [F_1][F_2] \neq 0 \).¶ This theorem implies in particular that the compact hyperbolic manifolds constructed by Gromov and Piatetski-Shapiro in [GP] have nontrivial virtual Betti numbers.