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The main purpose of this paper is to provide some criteria for the integrality and arithmeticity of a cocompact lattice corresponding to a complex ball of dimension two. Let G be a semi-simple Lie group of non-compact type and K be a maximal compact subgroup of G. G/K is then a symmetric space of non-compact type. Let Γ be a cocompact lattice of G. A lot of examples of such lattices are provided by arithmetic lattices (cf. [Bo1]). According to the Arithmeticity Theorem of Margulis [Mar], in the case that the symmetric space G/K has real rank at least 2, Γ has to be an arithmetic lattice. In the case that the symmetric space is either a quaternionic hyperbolic space or the Cayley hyperbolic plane, Corlette [C] establishes superrigidity of a lattice in those spaces, which together with the results of Gromov-Schoen [GS] established the arithmeticity of such lattices as well. For real hyperbolic spaces, there exist a lot of non-arithmetic lattices in each dimension as constructed by Gromov and Piatetski-Shapiro [GP]. According to the classification of semi-simple Lie groups, the only case remained is the class of complex hyperbolic manifolds of complex dimension at least 2. In this aspect, there are examples of non-arithmetic lattice in dimension 2 and 3 (cf. [Mos] [MD]). We do not know of examples in higher dimensions. In both real and complex hyperbolic cases, it is interesting to give criteria to characterize arithmetic lattices. It is well-known since the work of Margulis (cf. [Zi]) that arithmeticity follows from superrigidity. In [MSY], a general Bochner type formula was formulated, the linear version of which yields vanishing theorems including Matsushima’s Vanishing Theorem and Kazhdan’s Property T for such lattices, while the semi-linear version yields a uniform proof of the superrigidity results of Margulis and Corlette for cocompact lattices. In this way, problems about vanishing theorems of cohomology groups and superrigidity are put in a equal footing. For the complex or real hyperbolic cases, the Bochner formula does not lead to vanishing or rigidity properties as in the other cases. In fact, there are examples with non-vanishing first Betti numbers (cf. [BW]). However, it naturally leads us to look for criteria for arithmeticity of complex and real rank one cases in terms of restrictions on cohomology groups. A question known in this direction is the following which was attributed to Rogawski (cf. [Re]).

INTEGRALITY AND ARITHMETICITY OF CO-COMPACT LATTICE CORRESPONDING TO CERTAIN COMPLEX TWO-BALL QUOTIENTS OF PICARD NUMBER ONE