Homological Wideness, “Class Field Theory” for Covers, and a Number Theoretical Analogue

Abstract

AbstractIn a previous chapter, we have constructed a particular Riemannian covering realising a wreath product. In this chapter, we first return to that example and use class field theory for Riemannian coverings (à la Sunada) to study the behaviour of geodesic in such covers. We then relate, in the general case, homological wideness of a group G acting on a manifold M (i.e., the question whether the first homology of M contains the regular representation of G) to the existence of geodesics with certain splitting behaviour. In exact analogy to an classical argument in analytic number theory, we use the Ruelle zeta function to show the existence of infinitely many totally split geodesics for a given covering in the negative curvature case. Finally, the analogy with class field theory allows us to study an analogue of homological wideness in the theory of extensions of number fields.