  Let X' be the toroidal compactification of the quotient of the complex 2-ball
by a torsion free lattice G of SU(2,1). We say that X'is co-abelian if there is
an abelian surface, birational to X'. The present work can be viewed as an
illustration for the presence of a plenty of non-compact co-abelian torsion
free toroidal compactifications. More precisely, it shows that all the
admissible values for the volume of a torsion free quotient of the complex
2-ball are attained by co-abelian Picard modular ones over Eisenstein numbers.
The article provides three types of infinite series of finite unramified
coverings of co-abelian, torsion free, Picard modular toroidal
compactifications over Eisenstein numbers, with infinitely increasing volumes.
The first type is supported by mutually birational members with fixed number of
cusps. The second kind is with mutually birational terms and infinitely
increasing number of cusps. The third kind of series relates mutually
non-birational toroidal compactifications with infinitely increasing number of
cusps.
