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We study fundamental groups of toroidal compactifications of non compact ballquotients and show that the Shafarevich conjecture on holomorphic convexity forthese complex projective manifolds is satisfied in dimension 2 provided thecorresponding lattice is arithmetic and small enough. The method is to showthat the Albanese mapping on an {\'e}tale covering space generates jets on theinterior, if the lattice is small enough. We also explore some specificexamples of Picard-Eisenstein type.

Orbifold K{\"a}hler Groups related to arithmetic complex hyperbolic lattices