Zeta functions of discrete groups acting on trees

This paper generalizes Bass' work on zeta functions for uniform tree lattic es. Using the theory of von Neumann algebras, machinery is developed to def ine the zeta function of a discrete group of automorphisms of a bounded deg ree tree. The main theorems relate the zeta function to determinants of ope rators defined on edges or vertices of the tree. A zeta function associated to a non-uniform tree lattice with appropriate H ilbert representation is defined. Zeta functions are defined for infinite g raphs with a cocompact or finite covolume group action. (C) 2001 Academic P ress.