On the homology and the spectrum of hyperbolic manifolds (Q2709136)

This paper is a presentation of several results of the author, whose proof is going to be published in a paper announced in L'Enseignement Mathématique. It is quite remarkable that he constructs tools to find many examples of non-isometric isospectral hyperbolic manifolds, in particular he proves that such examples exist in any dimension. Notice that, by Mostow rigidity, those manifolds are not homeomorphic in dimension three or higher. He also develops techniques to show that certain examples of hyperbolic manifolds have infinite virtual first Betti number (i.e. the first Betti number of finite coverings is not bounded). NEWLINENEWLINENEWLINEAll these results are based on the study of geodesic cycles in hyperbolic manifolds. Those are cycles in homology represented by totally geodesic immersed submanifolds. The main idea is to lift those immersed cycles to embedded cycles in some finite covering, which is always possible if the ambient manifold has finite volume. In fact, a cycle of codimension one can be lifted to two nonseparating totally geodesic embedded submanifolds in some finite covering. This is the key theorem that allows to prove the results stated above, by cutting and pasting along those nonseparating submanifolds and by producing surjections of the fundamental group onto the free group of rank two. NEWLINENEWLINENEWLINEA family of examples is developed in detail, namely the hyperbolic 3-manifolds whose group has finite index in one of the eight reflection groups, called honeycombs by Coxeter. It is proved that those manifolds have a codimension one geodesic cycle, and therefore they have infinite virtual first Betti number and allow to construct many examples of non-isometric isospectral manifolds.NEWLINENEWLINEFor the entire collection see [Zbl 0955.00015].