Arithmeticity of complex hyperbolic triangle groups

TL;DR

This paper investigates the finiteness of complex hyperbolic triangle groups that form arithmetic lattices in PU(2, 1), extending classical results from Fuchsian groups to complex hyperbolic geometry.

Contribution

It establishes finiteness results for complex hyperbolic triangle groups with rational angular invariants, especially for right and equilateral triangles, generalizing known theorems.

Findings

Finitely many such groups with rational angular invariant form arithmetic lattices.

Finiteness results for right and equilateral triangle cases.

Extension of Takeuchi's finiteness theorem to complex hyperbolic groups.

Abstract

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.