Zeta Functions Of Discrete Groups Acting On Trees

1999/08/14 by Bryan Clair, Clair, Bryan, Shahriar Mokhtari-Sharghi +1 · 1 citation
Computer Science · Mathematics · #Advanced Operator Algebra Research #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Group Theory (math.GR) #Topological and Geometric Data Analysis #math.CO #math.GR

paper · pdf · doi:10.48550/arxiv.math/9908068

24 pages, 2 figures

arxiv created 1999/08/14 · openalex publication_date 1999/08/14 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

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• Chamber zeta function and closed galleries in the standard non-uniform complex from PGL₃

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