Selberg Zeta and Theta Functions

This volume provides a comprehensive exposition of the theory of Selberg zeta and theta functions for bundles on compact locally symmetric spaces of rank 1. The connection between these functions and the spectrum of certain elliptic differential operators is provided by a version of the Selberg trace formula. The theta function is a regularized trace of the wave group and originally defined geometrically, the Selberg zeta function has a representation in terms of regularized determinants. A description of its singularities and results are employed in order to establish a functional equation and further properties of the Ruelle zeta function, illustrated by examples. Additional chapters are devoted to the theta function of Riemannian surfaces with cusps and to alternative descriptions of the singularities of the Selberg zeta function in terms of Lie algebra and group cohomology. Topics discussed include: the process from the wave equation to the trace formula; singularities of the theta function; a determinant representation of the Selberg zeta function; the functional equations of Selberg and Ruelle zeta functions; and Dirac operators and form Laplacians.