  We present a search for galaxy clusters in the fields of three bona-fide
short GRBs (050709, 050724, and 051221a) and the putative short burst GRB
050911 using multi-slit optical spectroscopy. These observations are part of a
long-term program to constrain the progenitor age distribution based on the
fraction of short GRBs in galaxy clusters and early-type galaxies. We find no
evidence for cluster associations at the redshifts of the first three bursts,
but we confirm the presence of the cluster EDCC 493 within the error circle of
GRB 050911 and determine its redshift, z=0.1646, and velocity dispersion, ~660
km/s. In addition, our analysis of Swift/XRT observations of this burst reveals
diffuse X-ray emission coincident with the optical cluster position, with a
luminosity, 4.9e42 erg/s, and a temperature, kT=0.9 keV. The inferred mass of
the cluster is 2.5e13 Msun, and the probability of chance coincidence is about
0.1-1%, indicating an association with GRB 050911 at the 2.6-3.2 sigma
confidence level. A search for diffuse X-ray emission in coincidence with the
fifteen other short GRBs observed with XRT and Chandra reveals that with the
exception of the previously noted cluster ZwCl 1234.0+02916 likely associated
with GRB 050509b, no additional associations are evident to a typical limit of
3e-14 erg/s/cm^2, or M<5e13 Msun assuming a typical z=0.3. The resulting
fraction of short GRBs hosted by galaxy clusters of about 20% is in rough
agreement with the fraction of stellar mass in clusters of ~10-20%.
