Upper panel: spatial distribution of neutral hydrogen gas at T< 10⁵ K in the large-scale environment of the Local Group, as predicted from a constrained cosmological simulation from the CLUES project (N14). The color scale reflects the total hydrogen gas density, where the green filamentary structures outside the virial radii of the LG member galaxies lie in the density range n_H = 10^-5–10^-3 cm^-3. Positions and virial radii of Milky Way and M31 are indicated with white dashed (Milky Way) and solid (M31) circles. Lower panel: zoom-in of the gas distribution in and around the simulated MW/M31 galaxies and the bulk motions of that gas with respect to the Local Group barycenter in the x/y plane. The white arrows show the gas velocity field, with the largest ones representing a velocity of 130 km s^-1. The black arrows indicate the velocities of the MW and M31 galaxies. Their absolute space velocities are 67 and 76 km s^-1, respectively. The bright colors again indicate regions of enhanced hydrogen density.