H0LiCOW II. Spectroscopic survey and galaxy-group identification of the
  strong gravitational lens system HE0435-1223
  Galaxies located in the environment or on the line of sight towards
gravitational lenses can significantly affect lensing observables, and can lead
to systematic errors on the measurement of $H_0$ from the time-delay technique.
We present the results of a systematic spectroscopic identification of the
galaxies in the field of view of the lensed quasar HE0435-1223, using the W. M.
Keck, Gemini and ESO-Very Large telescopes. Our new catalog triples the number
of known galaxy redshifts in the vicinity of the lens, expanding to 102 the
number of measured redshifts for galaxies separated by less than 3 arcmin from
the lens. We complement our catalog with literature data to gather redshifts up
to 15 arcmin from the lens, and search for galaxy groups or clusters projected
towards HE0435-1223. We confirm that the lens is a member of a small group that
includes at least 12 galaxies, and find 8 other group candidates near the line
of sight of the lens. The flexion shift, namely the shift of lensed images
produced by high order perturbation of the lens potential, is calculated for
each galaxy/group and used to identify which objects produce the largest
perturbation of the lens potential. This analysis demonstrates that i) at most
three of the five brightest galaxies projected within 12 arcsec of the lens
need to be explicitly used in the lens models, and ii) the groups can be
treated in the lens model as an external tidal field (shear) contribution.
