VIRTUAL BETTI NUMBERS OF SOME HYPERBOLIC 3-MANIFOLDS

Abstract

This chapter discusses virtual Betti numbers of some hyperbolic 3-manifolds. The n-th virtual Betti number of a manifold means the supremum of the n-th Betti numbers of all its finite sheeted coverings. The first virtual Betti number of a hyperbolic 3-manifold is infinity. A 3-manifold with infinite fundamental group is finitely covered by a manifold with positive first Betti number. A monodromy map can be chosen to be covered by a homeomorphism on the universal cover, which commutes with the linear map via the developing map.