Arithmeticity of complex hyperbolic triangle groups

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