  This paper addresses strong cosmic censorship for spacetimes with
self-gravitating collisionless matter, evolving from surface-symmetric compact
initial data. The global dynamics exhibit qualitatively different features
according to the sign of the curvature $k$ of the symmetric surfaces and the
cosmological constant $\Lambda$. With a suitable formulation, the question of
strong cosmic censorship is settled in the affirmative if $\Lambda=0$ or
$k\le0$, $\Lambda>0$. In the case $\Lambda>0$, $k=1$, we give a detailed
geometric characterization of possible "boundary" components of spacetime; the
remaining obstruction to showing strong cosmic censorship in this case has to
do with the possible formation of extremal Schwarzschild-de Sitter-type black
holes. In the special case that the initial symmetric surfaces are all
expanding, strong cosmic censorship is shown in the past for all $k,\Lambda$.
Finally, our results also lead to a geometric characterization of the future
boundary of black hole interiors for the collapse of asymptotically flat data:
in particular, in the case of small perturbations of Schwarzschild data, it is
shown that these solutions do not exhibit Cauchy horizons emanating from $i^+$
with strictly positive limiting area radius.
