Toroidal compactifications of torsion free local complex hyperbolic spaces

Let B be the complex n-dimensional ball and X' be the toroidal compactification of a quotient B/G by a torsion free lattice G of SU(n,1). For an arbitrary G-rational boundary point p, denote by U(p) the commutant of the unipotent radical of the stabilizer of p in SU(n,1) and put G(U) for the subgroup of G, generated by the intersections of G with U(p) for all G-rational boundary points p. The present note establishes that the fundamental group of X' is isomorphic to the quotient G / G(U). As a consequence, the first integral homology group of X' turns to be a quotient of the first integral homology group of B/G by a finite group. The work shows that for any natural number N, there is a normal subgroup G(N) of G of finite index, such that the unramified covering of B/G by B/G(N), induced by the identity of the ball B extends to a covering of the corresponding toroidal compactifications with ramification index greater than N over the toroidal compactifying divisor of B/G(N). The argument exploits the residual finiteness of the lattices in SU(n,1). In the case of a complex dimension 2, the geometric genus of X' equals 1. If X' is not of general type, then the irregularity of X' does not exceed 2 and equals 2 only when X' is birational to an abelian surface. The torsion free surfaces X' of minimal volume are characterized by the Kodaira-Enriques classification types of their minimal models, as well as by lower and upper bounds on the number of the cusps.