We demonstrate the growth of low density anti-phase boundaries, crack-free GaAs epilayers, by Molecular Beam Epitaxy on silicon (001) substrates. The method relies on the deposition of thick GaAs on a suspended Ge buffer realized on top of deeply patterned Si substrates by means of a three-temperature procedure for the growth. This approach allows to suppress, at the same time, both threading dislocations and thermal strain in the epilayer and to remove anti-phase boundaries even in absence of substrate tilt. Photoluminescence measurements show the good uniformity and the high optical quality of AlGaAs/GaAs quantum well structures realized on top of such GaAs layer.