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The co-evolution between supermassive black holes and their environment is most directly traced by the hot atmospheres of dark matter halos. Cooling of the hot atmosphere supplies the central regions with fresh gas, igniting active galactic nuclei (AGN) with long duty cycles. Outflows from the central engine tightly couple with the surrounding gaseous medium and provide the dominant heating source preventing runaway cooling. Every major modern hydrodynamical simulation suite now includes a prescription for AGN feedback to reproduce realistic populations of galaxies. However, the mechanisms governing the feeding/feedback cycle between the central black holes and their surrounding galaxies and halos are still poorly understood. Galaxy groups are uniquely suited to constrain the mechanisms governing the cooling-heating balance, as the energy supplied by the central AGN can exceed the gravitational binding energy of halo gas particles. Here we provide a brief overview of our knowledge of the impact of AGN on the hot atmospheres of galaxy groups, with a specific focus on the thermodynamic profiles of groups. We then present our on-going efforts to improve on the implementation of AGN feedback in galaxy evolution models by providing precise benchmarks on the properties of galaxy groups. We introduce the ~ Group AGN Project (X-GAP), a large program on ~ targeting a sample of 49 galaxy groups out to R₅₀₀₂.
Eckert et al. (Mon,) studied this question.