The Anosov theorem for exponential solvmanifolds

Abstract

A well-known lower bound for the number of fixed points of a self-map / : X -> X is the Nielsen number N(f). Unfortunately, the Nielsen number is difficult to calculate. The Lefschetz number (/), on the other hand, is readily computable, but usually does not estimate the number of fixed points. It is known that N(f) = \L(f)\ for all maps on nilmanifolds (homogeneous spaces of nilpotent Lie groups) and that N(f) > \L(f)\ for all maps on solvmanifolds (homogeneous spaces of solvable Lie groups). Typically, though, the strict inequality holds, so the Nielsen number cannot be completely computed from the Lefschetz number. In the present work, we produce a large class of solvmanifolds for which N(f) = \L(f)\ for all self maps. This class includes exponential solvmanifolds: solvmanifolds for which the corresponding exponential map is surjective. Our methods provide Nielsen and Lefschetz number product theorems for the Mostow fibrations of these solvmanifolds, even though the maps on the fibers in general will belong to varying homotopy classes.