(abridged) We introduce our survey of galaxy groups at 0.85<z<1, as an extension of the Group Environment and Evolution Collaboration (GEEC). Here we present the first results, based on Gemini GMOS-S nod-and-shuffle spectroscopy of seven galaxy groups selected from spectroscopically confirmed, extended XMM detections in COSMOS. In total we have over 100 confirmed group members, and four of the groups have >15 members. The dynamical mass estimates are in good agreement with the masses estimated from the X-ray luminosity, with most of the groups having 13<log(Mdyn/Msun)<14. Our spectroscopic sample is statistically complete for all galaxies with Mstar>1E10.1 Msun, and for blue galaxies we sample masses as low as Mstar=1E8.8 Msun. Like lower-redshift groups, these systems are dominated by red galaxies, at all stellar masses Mstar>1E10.1 Msun. Few group galaxies inhabit the blue cloud that dominates the surrounding field; instead, we find a large and possibly distinct population of galaxies with intermediate colours. The green valley that exists at low redshift is instead well-populated in these groups, containing ~30 per cent of galaxies. These do not appear to be exceptionally dusty galaxies, and about half show prominent Balmer-absorption lines. Furthermore, their HST morphologies appear to be intermediate between those of red-sequence and blue-cloud galaxies of the same stellar mass. We postulate that these are a transient population, migrating from the blue cloud to the red sequence, with a star formation rate that declines with an exponential timescale 0.6 Gyr< tau < 2 Gyr. Their prominence among the group galaxy population, and the marked lack of blue, star-forming galaxies, provides evidence that the group environment either directly reduces star formation in member galaxies, or at least prevents its rejuvenation during the normal cycle of galaxy evolution.