  [Abridged] We study the interface between galaxies and the intergalactic
medium by measuring the absorption by neutral hydrogen in the vicinity of
star-forming galaxies at z~2.4. Our sample consists of 679 rest-frame-UV
selected galaxies with spectroscopic redshifts that have impact parameters < 2
(proper) Mpc to the line of sight of one of 15 bright, background QSOs. We
present the first 2-D maps of the absorption around galaxies, plotting the
median Ly-a pixel optical depth as a function of transverse and line of sight,
LOS, separation from galaxies. The median optical depth, and hence the median
density of atomic hydrogen, drops by more than an order of magnitude around 100
kpc, which is similar to the virial radius of the halos thought to host the
galaxies. The median remains enhanced (>3 sigma level) out to at least 2.8 Mpc
(i.e. > 9 comoving Mpc). Within 100 (200) kpc, and over \pm 165 km/s, the
covering fraction of gas with Ly-a optical depth greater than unity is
100(+0)(-32)% (86(+14)(-18)%). Absorbers with tau(Ly-a)> 0.1 are typically
closer to galaxies than random. The mean galaxy overdensity around absorbers
increases with the optical depth and also as the length scale over which the
galaxy overdensity is evaluated is decreased. Absorbers with tau(Ly-a)~1 reside
in regions where the galaxy number density is close to the cosmic mean on
scales > 0.25 Mpc. We detect two types of redshift space anisotropies. On
scales < 200 km/s, or < 1 Mpc, the absorption is stronger along the LOS than in
the transverse direction. This "finger of God" effect may be partly due to
redshift errors, but is probably dominated by gas motions within or very close
to the halos. On the other hand, on scales of 1.4 - 2.0 Mpc the absorption is
compressed along the LOS (>3 sigma significance), which we attribute to
large-scale infall (i.e. the Kaiser effect).
