Zeta Functions of Discrete Groups Acting on Trees

This paper generalizes Bass’ work on zeta functions for uniform tree lattices. Using the theory of von Neumann algebras, machinery is developed to define the zeta function of a discrete group of automorphisms of a bounded degree tree. The main theorems relate the zeta function to determinants of operators defined on edges or vertices of the tree. A zeta function associated to a non-uniform tree lattice with appropriate Hilbert representation is defined. Zeta functions are defined for infinite graphs with a cocompact or finite covolume group action. Introduction The zeta function associated to a finite graph originated in work of Ihara [11, 10], which proves a structure theorem for torsion-free discrete cocompact subgroups of PGL(2, kp) (where kp is a p-adic number field or a field of power series over a finite constant field). If Γ < PGL(2, kp) is such a group, Ihara shows that Γ is in fact a free group and defines a zeta function associated to Γ. An element 1 != γ ∈ Γ is called primitive if it generates its centralizer in Γ. Let P(Γ) denote the set of conjugacy classes of primitive elements of Γ. If λ1 and λ2 are the eigenvalues of a representative of γ ∈ PGL(2, kp), define l(γ) = |vp(λ1λ −1 2 )|, where vp is the normalized valuation for kp. Ihara defines the zeta function attached to Γ as: ZΓ(u) = ∏