Abstract
Let B ⊂ C be the unit ball and Γ be a lattice of SU(2, 1). Bearing in mind that all compact Riemann surfaces are discrete quotients of the unit disc ∆ ⊂ C, Holzapfel conjectures that the discrete ball quotients B/Γ and their compactifications are widely spread among the smooth projective surfaces. There are known ball quotients B/Γ of general type, as well as rational, abelian, K3 and elliptic ones. The present note constructs three noncompact ball quotients, which are birational, respectively, to a hyperelliptic, Enriques or a ruled surface with an elliptic base. As a result, we establish that the ball quotient surfaces have representatives in any of the eight Enriques classification classes of smooth projective surfaces.
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