  Complex hyperbolic triangle groups were first considered by Mostow in
building the first nonarithmetic lattices in PU(2, 1). They are a natural
generalization of the classical triangle groups acting on the hyperbolic plane.
A well-known theorem of Takeuchi is that there are only finitely many Fuchsian
triangle groups that are also an arithmetic lattice in PSL_2(R). We consider
similar finiteness theorem for complex hyperbolic triangle groups. In
particular, we show that there are finitely many complex hyperbolic triangle
groups with rational angular invariant which determine an arithmetic lattice in
PU(2, 1). We also prove finiteness when the triangle is a right or equilateral
triangle, the latter case being the one which has attracted the greatest amount
of attention since Mostow's original work.
