Arithmeticity of complex hyperbolic triangle groups

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 PSL 2.(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). © 2012 by Pacific Journal of Mathematics.