NGC 4936: abundant multi-phase medium in the central weak X-ray cool core and optically dark H I clouds in the group

We present a multiwavelength analysis of NGC 4936, the central galaxy of a weak cool-core group hosting an unusually large cold/warm gas reservoir. NGC 4936 contains sim2times10^8 M_odot of molecular gas in a rotating disk of radius sim1 kpc, but no detected H I, with a 3σ upper limit of 1.13times10^7 M_odot within sim2.5 kpc. The resulting H_2/H I mass ratio >17 is consistent with a predominantly molecular medium in the high-pressure environments of central galaxies. SOAR imaging and MUSE spectroscopy reveal ionised gas extending to sim15 kpc in a complex filamentary morphology that broadly follows the molecular gas kinematics while extending well beyond the CO disk. Dust filaments similarly extend to sim14 kpc and appear to interact with a companion galaxy located only sim2.5 kpc away in projection. NGC 4936 has a star formation rate of 0.29 M_odot yr^{-1}, and its AGN shows sub-mm and X-ray variability. We favour atmospheric condensation as the primary origin of the molecular gas, although externally acquired gas may contribute. Our recalculated thermodynamic profiles reach t_{rm cool}/t_{rm ff}sim25 in the innermost region, placing the system near the threshold for multiphase condensation. The ongoing interaction may perturb the atmosphere and promote condensation while shaping the morphology and kinematics of the multiphase medium, rather than directly supplying most of the central cold gas. We additionally detect 20 H I structures in the surrounding group environment, with masses of sim10^7--10^8 M_odot, including several with no obvious optical counterparts.