  In the SSA22 field which exhibits a large-scale proto-cluster at $z=3.1$, we
carried out a spectroscopic survey for Lyman Break Galaxies (LBGs) with the
VLT/VIMOS and identified 78 confident LBGs at $z=2.5$--4. We stacked their
spectra in the observer's frame by using a sophisticated method. Analyzing the
composite spectrum, we have revealed that the large-scale proto-cluster at
$z=3.1$ has a strong HI absorption dip of rest-frame equivalent width of $-1.7$
A. Another strong absorption dip found at $z=3.28$ is associated with a
modestly high-density LBG peak, similar to that at $z=3.1$. We have also
detected an HI transparency peak at $z=2.98$ in the composite spectrum,
coincident with a void in the LBG distribution. In this paper, we also
investigated the relation between LBGs, HI gas and AGNs at $z=3$--4 in the
SSA22 field. Two AGNs at $z=3.353$ and 3.801 are, respectively, associated with
the LBG concentration of an overdensity factor $\delta_{\rm LBG}\simeq2$ in the
present statistics. Another structure at $z=3.453$ is remarkable: 20 comoving
Mpc scale dense HI gas which is not associated with any apparent LBG
overdensity but involving a pair of AGNs. Such structure may be a new type of
the AGN-matter correlation. If the inhomogeneous structures over a comoving Gpc
scale found in this paper are confirmed with sufficient statistics in the
future, the SSA22 field will become a key region to test the standard cold dark
matter structure formation scenario.
