  Recently, relations connecting the SMBH mass of central galaxies and global
properties of the hosting cluster, such as temperature and mass, were observed.
We investigate the correlation between SMBH mass and cluster mass and
temperature, their establishment and evolution. We compare their scatter to
that of the classical $M_{\rm BH}-M_{\rm BCG}$ relation. We study how gas
accretion and BH-BH mergers contribute to SMBH growth across cosmic time. We
employed 135 groups and clusters with a mass range $1.4\times
10^{13}M_{\odot}-2.5\times 10^{15} M_{\odot}$ extracted from a set of 29
zoom-in cosmological hydro-dynamical simulations where the baryonic physics is
treated with various sub-grid models, including feedback by AGN. In our
simulations we find that $M_{\rm BH}$ correlates well with $M_{500}$ and
$T_{500}$, with the scatter around these relations compatible within $2\sigma$
with the scatter around $M_{\rm BH}-M_{\rm BCG}$ at $z=0$. The $M_{\rm
BH}-M_{500}$ relation evolves with time, becoming shallower at lower redshift
as a direct consequence of hierarchical structure formation. On average, in our
simulations the contribution of gas accretion to the total SMBH mass dominates
for the majority of the cosmic time ($z>0.4$), while in the last 2 Gyr the
BH-BH mergers become a larger contributor. During this last process,
substructures hosting SMBHs are disrupted in the merger process with the BCG
and the unbound stars enrich the diffuse stellar component rather than increase
BCG mass. From the results obtained in our simulations with simple sub-grid
models we conclude that the scatter around the $M_{\rm BH}-T_{500}$ relation is
comparable to the scatter around the $M_{\rm BH}-M_{\rm BCG}$ relation and
that, given the observational difficulties related to the estimation of the BCG
mass, clusters temperature and mass can be a useful proxy for the SMBHs mass,
especially at high redshift.
