The dynamics of the Local Group and its environment provide a unique
challenge to cosmological models. The velocity field within 5h-1 Mpc of the
Local Group (LG) is extremely ``cold''. The deviation from a pure Hubble flow,
characterized by the observed radial peculiar velocity dispersion, is measured
to be about 60km/s. We compare the local velocity field with similarly defined
regions extracted from N-body simulations of Universes dominated by cold dark
matter (CDM). This test is able to strongly discriminate between models that
have different mean mass densities. We find that neither the Omega=1 (SCDM) nor
Omega=0.3 (OCDM) cold dark matter models can produce a single candidate Local
Group that is embedded in a region with such small peculiar velocities. For
these models, we measure velocity dispersions between 500-700km/s and
150-300km/s respectively, more than twice the observed value. Although both CDM
models fail to produce environments similar to those of our Local Group on a
scale of a few Mpc, they can give rise to many binary systems that have similar
orbital properties as the Milky Way--Andromeda system. The local,
gravitationally induced bias of halos in the CDM ``Local Group'' environment,
if defined within a sphere of 10 Mpc around each Local Group is about 1.5,
independent of Omega. No biasing scheme could reconcile the measured velocity
dispersions around Local Groups with the observed one. Identification of binary
systems using a halo finder (named Skid
(http://www-hpcc.astro.washington.edu/tools/DENMAX for a public version)) based
on local density maxima instead of a simple linking algorithm, gives a much
more complete sample. We show that a standard ``friend of friends'' algorithm
would miss 40% of the LG candidates present in the simulations.Comment: Latex file (19 pages) + 13 figures. Submitted to New Astronomy. Two
MPEG movies were not included. Also available (this time with the movies) at
http://www-hpcc.astro.washington.edu/faculty/fabio/index.htm
