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 100 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 cluster 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. The statistical impact of the groups and voids on the lens model is presented in a companion paper H0LiCOW III. The exhaustive lens modeling of HE0435-1223, used for cosmological inference, including all the environmental sources of systematic errors, is presented in another companion paper H0LiCOW IV.