A Spectroscopic Survey of the Fields of 28 Strong Gravitational Lenses:
  Implications for $H_0$
  Strong gravitational lensing provides an independent measurement of the
Hubble parameter ($H_0$). One remaining systematic is a bias from the
additional mass due to a galaxy group at the lens redshift or along the
sightline. We quantify this bias for more than 20 strong lenses that have
well-sampled sightline mass distributions, focusing on the convergence $\kappa$
and shear $\gamma$. In 23% of these fields, a lens group contributes a $\ge$1%
convergence bias; in 57%, there is a similarly significant line-of-sight group.
For the nine time delay lens systems, $H_0$ is overestimated by 11$^{+3}_{-2}$%
on average when groups are ignored. In 67% of fields with total $\kappa \ge$
0.01, line-of-sight groups contribute $\gtrsim 2\times$ more convergence than
do lens groups, indicating that the lens group is not the only important mass.
Lens environment affects the ratio of four (quad) to two (double) image
systems; all seven quads have lens groups while only three of 10 doubles do,
and the highest convergences due to lens groups are in quads. We calibrate the
$\gamma$-$\kappa$ relation: $\log(\kappa_{\rm{tot}}) = (1.94 \pm 0.34)
\log(\gamma_{\rm{tot}}) + (1.31 \pm 0.49)$ with a rms scatter of 0.34 dex.
Shear, which, unlike convergence, can be measured directly from lensed images,
can be a poor predictor of $\kappa$; for 19% of our fields, $\kappa$ is
$\gtrsim 2\gamma$. Thus, accurate cosmology using strong gravitational lenses
requires precise measurement and correction for all significant structures in
each lens field.
