  Overdense regions at high redshift ($z \gtrsim 2$) are perfect laboratories
to study the relations between environment and SMBH growth, and the AGN
feedback processes on the surrounding galaxies and diffuse gas. In this white
paper, we discuss how AXIS will 1) constrain the AGN incidence in
protoclusters, as a function of parameters such as redshift, overdensity, mass
of the structure; 2) search for low-luminosity and obscured AGN in the
satellite galaxies of luminous QSOs at $z>6$, exploiting the large galaxy
density around such biased objects; 3) probe the AGN feedback on the proto-ICM
via the measurement of the AGN contribution to the gas ionization and
excitation, and the detection of extended X-ray emission from the ionized gas
and from radio jets; 4) discover new large-scale structures in the wide and
deep AXIS surveys as spikes in the redshift distribution of X-ray sources.
These goals can be achieved only with an X-ray mission with the capabilities of
AXIS, ensuring a strong synergy with current and future state-of-the-art
facilities in other wavelengths. This White Paper is part of a series
commissioned for the AXIS Probe Concept Mission; additional AXIS White Papers
can be found at http://axis.astro.umd.edu/ with a mission overview at
https://arxiv.org/abs/2311.00780.
