  Supermassive black holes (BHs) residing in the brightest cluster galaxies are
over-massive relative to the stellar bulge mass or central stellar velocity
dispersion of their host galaxies. As BHs residing at the bottom of the galaxy
cluster's potential well may undergo physical processes that are driven by the
large-scale characteristics of the galaxy clusters, it is possible that the
growth of these BHs is (indirectly) governed by the properties of their host
clusters. In this work, we explore the connection between the mass of BHs
residing in the brightest group/cluster galaxies (BGGs/BCGs) and the virial
temperature, and hence total gravitating mass, of galaxy groups/clusters. To
this end, we investigate a sample of 17 BGGs/BCGs with dynamical BH mass
measurements and utilize XMM-Newton X-ray observations to measure the virial
temperatures and infer the $M_{\rm 500}$ mass of the galaxy groups/clusters. We
find that the $M_{\rm BH} - kT$ relation is significantly tighter and exhibits
smaller scatter than the $M_{\rm BH} - M_{\rm bulge}$ relations. The
best-fitting power-law relations are $ \log_{10} (M_{\rm BH}/10^{9} \
\rm{M_{\odot}}) = 0.20 + 1.74 \log_{10} (kT/1 \ \rm{keV}) $ and $ \log_{10}
(M_{\rm BH}/10^{9} \ \rm{M_{\odot}}) = -0.80 + 1.72 \log_{10} (M_{\rm
bulge}/10^{11} \ M_{\odot})$. Thus, the BH mass of BGGs/BCGs may be set by
physical processes that are governed by the properties of the host galaxy
group/cluster. These results are confronted with the Horizon-AGN simulation,
which reproduces the observed relations well, albeit the simulated relations
exhibit notably smaller scatter.
