Integrality and arithmeticity of co-compact lattice corresponding to certain complex two-ball quotients of Picard number one
type: Item
instance of: scholarly article
title: Integrality and arithmeticity of co-compact lattice corresponding to certain complex two-ball quotients of Picard number one
zbMATH Open document ID: 1088.22007
published in: The Asian Journal of Mathematics
publication date: 2005-03-21T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ajm/1087840911
review text: Let \(G\) be a semi-simple Lie group of non-compact type and \(K\) be a maximal compact subgroup of \(G\), then \(G/K\) is a symmetric space of noncompact type. Let \(\Gamma\) be a co-compact lattice in \(G\). In the case when \(G/K\) has real rank greater than or equal to 2, \(\Gamma\) has to be an arithmetic lattice by Margulis' arithmeticity theorem. If \(G/K\) is either a quaternionic hyperbolic space or the Cayley hyperbolic plane, \(\Gamma\) is arithmetic by the results of Corlette and Gromov-Schoen. On the other hand, in the case of \(G/K\) being a real or a complex hyperbolic space non-arithmetic lattices were constructed by Gromov and Piatetski-Shapiro, and Deligne and Mostow. In these cases one might ask whether there are additional conditions that ensure the arithmeticity of the lattices. For lattices in \(PU(2,1)\) (acting on the complex hyperbolic plane), such a condition in terms of cohomology groups was conjectured by Rogawski. In Part A of the main theorem the author of the paper under review proves that lattices in \(PU(2,1)\) that satisfy the conditions conjectured by Rogawski are integral, which is slightly weaker than being arithmetic. In Parts B and C of the main theorem the author shows that under additional conditions on either the canonical line bundle or the absence of an immersed totally geodesic curve the lattices are arithmetic.
reviewed by: L. Lifschitz
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 32Q45
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 2148244
author: Sai-Kee Yeung
MaRDI profile type: Publication
DOI: 10.4310/AJM.2004.V8.N1.A9
description: scientific article; zbMATH DE number 2148244