Achieving an observational determination and a theoretical understanding of the Cosmic Star Formation History (CSFH) of the Universe is one of the biggest challenges in the study of galaxy formation and evolution. The most striking feature of the CSFH is a dramatic drop of the star formation rate, since z~1 to the present day, after a rather constant phase of high activity at z~1-2 or above. A quenching process is required to trigger such an abrupt decrease of the Star Formation (SF) activity. As a matter of fact, the progressive decline of the SF activity of the Universe since z~1 anti-correlates with the late-time increase of the number density of group-sized halos. It is, then, mandatory to follow this tantalizing lead and to ask if the very same process of assembly and growth of structures may be the main cause, or one of the major drivers, of the 1 order of magnitude decline in the CSFH. I will tackle this issue by showing that group galaxies evolve much faster than galaxies in low density regions. Star forming group galaxies, in particular, are perfectly on the Main Sequence (MS) at z~1 whereas at lower redshift they are quenched, thus, dropping off the MS quicker than field galaxies towards the region of SF quiescence. I will also show that the efficiency of the quenching process depends on the halo mass, being more efficient in the most massive halos, and thus, in most massive galaxies that tend to reside in such halos.