The Selberg trace formula and the Selberg zeta-function for cocompact discrete subgroups of \(SO_ +(1,n)\)
type: Item
instance of: scholarly article
title: The Selberg trace formula and the Selberg zeta-function for cocompact discrete subgroups of \(SO_ +(1,n)\)
zbMATH Open document ID: 0846.11033
DOI: 10.1007/BF01075633
author: L. B. Parnovskij
publication date: 1994-07-06T00:00:00Z
review text: Selberg's trace formula is derived for \(n\)-dimensional hyperbolic space in the cocompact case. The author follows the classical setup. Most parts of the calculations are omitted. Choosing a suitable test-function in the trace formula, the resulting loxodromic term is shown to be a logarithmic derivative of a meromorphic function which is then defined to be Selberg's zeta-function. For low dimensions this coincides with the usual definition. Analytic properties such as a functional equation are derived from the trace formula; a prime geodesic theorem follows in a standard way.
reviewed by: Roland Matthes
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 11M36
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 571845
zbMATH Keywords: Selberg's trace formula
zbMATH Keywords: hyperbolic space
zbMATH Keywords: Selberg's zeta-function
zbMATH Keywords: functional equation
zbMATH Keywords: prime geodesic theorem
MaRDI profile type: Publication
cites work: Q3237356
cites work: Zeta functions of Selberg's type for compact space forms of symmetric spaces of rank one
cites work: Discontinuous groups on three-dimensional hyperbolic space: analytical theory and arithmetic applications
cites work: Q3734489
cites work: Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen
Recommended article: Q3734489
Recommended article: The Selberg trace formula and Selberg zeta-function for cofinite Kleinian groups with finite-dimensional unitary representations
Recommended article: THE MEAN SQUARE OF THE LOGARITHMIC DERIVATIVE OF THE SELBERG ZETA FUNCTION FOR COCOMPACT DISCRETE SUBGROUPS
Recommended article: The Selberg zeta function and a local trace formula for Kleinian groups.
Recommended article: Q3778817
Recommended article: The Selberg trace formula for the Picard group SL(2, Z[i])
Recommended article: The Selberg zeta function for convex co-compact Schottky groups
Recommended article: An approach to the Selberg trace formula via the Selberg zeta-function
Recommended article: Q4855504
Recommended article: Q3491823
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 571845