Arithmeticity of complex hyperbolic triangle groups

Matthew Stover

Vol. 257 (2012), No. 1, 243–256

DOI: 10.2140/pjm.2012.257.243

Abstract

Complex hyperbolic triangle groups, originally studied by Mostow in building the first nonarithmetic lattices in PU(2,1), are a natural generalization of the classical triangle groups. A theorem of Takeuchi states that there are only finitely many Fuchsian triangle groups that determine an arithmetic lattice in PSL₂(ℝ), so triangle groups are generically nonarithmetic. We prove similar finiteness theorems for complex hyperbolic triangle groups that determine an arithmetic lattice in PU(2,1).

Keywords

complex hyperbolic geometry, arithmetic lattices, complex hyperbolic triangle groups

Mathematical Subject Classification 2010

Primary: 11F06, 20H10, 22E40

Milestones

Received: 12 October 2011

Revised: 30 November 2011

Accepted: 5 December 2011

Published: 19 June 2012