  We present Hubble Space Telescope Cosmic Origin Spectrograph (COS) UV line
spectroscopy and integral-field unit (IFU) observations of the intra-group
medium in Stephan's Quintet (SQ). SQ hosts a 30 kpc long shocked ridge
triggered by a galaxy collision at a relative velocity of 1000 km/s, where
large amounts of molecular gas coexist with a hot, X-ray emitting, plasma. COS
spectroscopy at five positions sampling the diverse environments of the SQ
intra-group medium reveals very broad (2000 km/s) Ly$\alpha$ line emission with
complex line shapes. The Ly$\alpha$ line profiles are similar to or much
broader than those of H$\beta$, [CII]$\lambda157.7\mu$m and CO~(1-0) emission.
The extreme breadth of the Ly$\alpha$ emission, compared with H$\beta$, implies
resonance scattering within the observed structure. Scattering indicates that
the neutral gas of the intra-group medium is clumpy, with a significant surface
covering factor. We observe significant variations in the Ly$\alpha$/H$\beta$
flux ratio between positions and velocity components. From the mean line ratio
averaged over positions and velocities, we estimate the effective escape
fraction of Ly$\alpha$ photons to be 10-30%. Remarkably, over more than four
orders of magnitude in temperature, the powers radiated by X-rays, Ly$\alpha$,
H$_2$, [CII] are comparable within a factor of a few, assuming that the ratio
of the Ly$\alpha$ to H$_2$ fluxes over the whole shocked intra-group medium
stay in line with those observed at those five positions. Both shocks and
mixing layers could contribute to the energy dissipation associated with a
turbulent energy cascade. Our results may be relevant for the cooling of gas at
high redshifts, where the metal content is lower than in this local system, and
a high amplitude of turbulence is more common.
