The Picard group, closed geodesics and zeta functions

Abstract

In this article we consider the Picard group ${\text {SL}}(2,\mathbb {Z}[i])$, viewed as a discrete subgroup of the isometries of hyperbolic space. We fix a canonical choice of generators and then construct a Markov partition for the action of the group on the sphere at infinity. Our main application is to the study of the zeta function associated to the associated three-dimensional hyperbolic manifold.