Abstract
We prove that for a compact complex hyperbolic space form of complex dimension two whose associated lattice is nonintegral, the first Betti number is virtually positive. This provides support to a question of Borel in favor of a conjecture of Thurston on virtual positivity of the first Betti number on complex hyperbolic space forms. The result can also be considered as a topological sufficient condition for the integrality of a lattice in PU(2,1).
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