  While many strong gravitational lens galaxies are suspected to lie in groups
or clusters of galaxies, environmental effects in lens models are often
unconstrained and sometimes ignored. We show that this creates significant
biases in a variety of lensing applications, by creating mock lenses associated
with each of 13 galaxies in a realistic model group, and then analyzing them
with standard techniques. We find that standard models of double lenses, which
neglect environment, grossly overestimate the ellipticity of the lens galaxy
(de/e~0.5) and the Hubble constant (dh/h~0.22). Standard models of quad lenses,
which approximate the environment as a tidal shear, recover the ellipticity
reasonably well (|de/e|<~0.24) but overestimate the Hubble constant
(dh/h~0.15), and have significant (~30%) errors in the millilensing analyses
used to constrain the amount of substructure in dark matter halos. For both
doubles and quads, standard models slightly overestimate the velocity
dispersion of the lens galaxy (d(sigma)/sigma~0.06), and underestimate the
magnifications of the images (d(mu)/mu ~ -0.25). Standard analyses of lens
statistics overestimate Omega_Lambda (by 0.05-0.14), and underestimate the
ratio of quads to doubles (by a factor of 2). These biases help explain some
long-standing puzzles (such as the high observed quad/double ratio), but
aggravate others (such as the low value of H_0 inferred from lensing). Most of
the biases are caused by neglect of the convergence from the mass associated
with the environment, but additional uncertainty is introduced by neglect of
higher-order terms. Fortunately, we show that directly observing and modeling
lens environments should make it possible to remove the biases and reduce the
uncertainties associated with environments to the few percent level. (Abridged)
