  In this paper we examine the relationship between the length spectrum and the
geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M. In
particular we analyze the extent to which the geometry of M is determined by
the closed geodesics coming from finite area totally geodesic surfaces. Using a
variety of techniques from analytic number theory, we address the following
problems: Is the commensurability class of an arithmetic hyperbolic 3-orbifold
determined by the lengths of closed geodesics lying on totally geodesic
surfaces?, Do there exist arithmetic hyperbolic 3-orbifolds whose "short"
geodesics do not lie on any totally geodesic surfaces?, and Do there exist
arithmetic hyperbolic 3-orbifolds whose "short" geodesics come from distinct
totally geodesic surfaces?
