Selberg's trace formula on the \(k\)-regular tree and applications
type: Item
instance of: scholarly article
title: Selberg's trace formula on the \(k\)-regular tree and applications
zbMATH Open document ID: 1083.11032
DOI: 10.1155/S016117120311126X
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2003-03-17T00:00:00Z
full work available at URL: https://eudml.org/doc/52172
review text: Summary: We survey graph theoretic analogues of the Selberg trace and pretrace formulas along with some applications. This paper includes a review of the basic geometry of a \(k\)-regular tree \(\Xi\) (symmetry group, geodesics, horocycles, and the analogue of the Laplace operator). A detailed discussion of the spherical functions is given. The spherical and horocycle transforms are considered (along with three basic examples, which may be viewed as a short table of these transforms). Two versions of the pretrace formula for a finite connected \(k\)-regular graph \(X\cong\Gamma\backslash\Xi\) are given along with two applications. The first application is to obtain an asymptotic formula for the number of closed paths of length \(r\) in \(X\) (without backtracking but possibly with tails). The second application is to deduce the chaotic properties of the induced geodesic flow on \(X\) (which is analogous to a result of Wallace for a compact quotient of the Poincaré upper half plane). Finally, the Selberg trace formula is deduced and applied to the Ihara zeta function of \(X\), leading to a graph theoretic analogue of the prime number theorem.
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 05C50
Mathematics Subject Classification ID: 43A90
zbMATH DE Number: 1883682
zbMATH Keywords: spherical functions
zbMATH Keywords: pretrace formula
zbMATH Keywords: Selberg trace formula
zbMATH Keywords: Ihara zeta function
author: Audrey A. Terras
author: Dorothy Wallace
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1883682