Galaxies grow in mass by accreting matter flowing along streams and filaments of the cosmic web. But they would become too massive, if there was not a super-massive black hole in their nucleus to inject energy and matter back into the intergalactic medium. The details of such a feedback mechanism are however still mysterious. Giant galaxies in cluster centers (BCGs) are a unique example of Black Hole/Intergalactic gas interaction. The goal of LYRICS is to understand the life cycle of gas in the presence of AGN-feedback by studying the large filamentary network surrounding those galaxies. LYRICS builds up on i) large datasets and archival observations (MUSE, ALMA, NOEMA) ii) new implementation of physical models of gas excitation and iii) mature hydrodynamical simulations including AGN-feedback. Organised around a very complementary team, LYRICS will provide the very first comprehensive picture of the origin and state of the gas circulating around BCGs.