  We present Giant Metrewave Radio Telescope 21-cm HI observations towards a
sample of compact radio sources behind galaxy groups, to search for cool HI.
The results -- from high dynamic range spectra for 8 lines-of-sight through 7
galaxy groups -- do not show any evidence for absorption by cool HI. At a
resolution of 20 km/s, the optical depth upper limits obtained were between
0.0075 and 0.035 (3sigma); these correspond to upper limits of a few times
10**23 m**-2 for the column density of any cool HI along these lines of sight
(assuming a spin temperature of 100 K).
