  The origin of the cold gas in central galaxies in groups is still a matter of
debate. We present Multi-Unit Spectroscopic Explorer (MUSE) observations of 18
optically selected local Brightest Group Galaxies (BGGs) to study the
kinematics and distribution of the optical emission-line gas. MUSE observations
reveal a distribution of gas morphologies including ten complex networks of
filaments extending up to 10 kpc to two compact (<3 kpc) and five extended (>5
kpc) disk-dominated structures. Some rotating disks show rings and elongated
structures arising from the central disk. The kinematics of the stellar
component is mainly rotation-dominated, which is very different from the
disturbed kinematics and distribution found in the filamentary sources. The
ionized gas is kinematically decoupled from the stellar component for most
systems, suggesting an external origin for the gas. We also find that the
Halpha luminosity correlates with the cold molecular mass. By exploring the
thermodynamical properties of the hot atmospheres, we find that the filamentary
sources and compact disks are found in systems with small central entropy
values and tcool/teddy ratios. This suggests that, like for Brightest Cluster
Galaxies in cool core clusters, the ionized gas are likely formed from hot halo
gas condensations, consistently with the Chaotic Cold Accretion simulations (as
shown via the C-ratio, Tat, and k-plot). We note that gaseous rotating disks
are more frequent than in BCGs. An explanation for the origin of the gas in
those objects is a contribution to gas fueling by mergers or group satellites,
as qualitatively hinted by some sources of the present sample. Nonetheless, we
discuss the possibility that some extended disks could also be a transition
stage in an evolutionary sequence including filaments, extended disks and
compact disks, as described by hot gas condensation models of cooling flows.
