Abstract: Let S be a orientable surface of negative Euler characteristic. We
will discuss the action of pseudo-Anosov mapping classes of S on the homology
of various finite covers of S to which they lift. We will be particularly
interested in finding a lift of each pseudo-Anosov mapping class for which the
homological spectral radius is greater than one. For a given mapping class, we
will relate the study of this problem to the topology of the mapping torus.