  Recently many people have discussed the possibility that the universe is
hyperbolic and was in an inflationary phase in the early stage. Under these
assumptions, it is shown that the universe cannot have compact hyperbolic
time-slices. Though the universal covering space of the universe has a past
Cauchy horizon and can be extended analytically beyond it, the extended region
has densely many points which correspond to singularities of the compact
universe. The result is essentially attributed to the ergodicity of the
geodesic flow on a compact negatively curved manifold. Validity of the result
is also discussed in the case of inhomogeneous universe. Relationship with the
strong cosmic censorship conjecture is also discussed.
