The Selberg Zeta Function: A golden thread through hyperbolic geometry, dynamics and number theory

This thesis deals with the profound relationship that exists between the dynamics on surfaces of negative curvature, some spectral aspects of hyperbolic geometry, and the ergodic properties of continued fractions. One of the most elegant results to reveal the connections between the geodesic flow on a surface of constant negative curvature and the spectrum of the Laplace-Beltrami operator is the celebrated trace formula of Selberg. From a standpoint which is not that of this thesis, one could say that any such trace formula represents one of the most direct connections between observations of classical dynamics and observations of quantum dynamics. At any rate, the main mathematical tool behind the original trace formula is the Selberg zeta function. The purpose of the thesis is to review certain papers of D. H. Mayer, and more recently of C.Bonanno and S. Isola, on certain representations of the Selberg zeta function for surfaces of constant negative curvature based on families of transfer operators for the Gauss and Farey maps, which are the dynamical systems associated to the continued-fraction expansion of real numbers. In particular, we extend some results of Bonanno and Isola to subgroups of the full modular group PSL(2;Z)