If a group $G$ acts properly and cocompactly by isometries on
a simply-connected space of non-positive
curvature then one can say a great deal about the
structure of $G$. Some of the most significant
properties of the groups $G$ that admit these cocompact actions
are inherited by subgroups $\G\subset G$,
but many others are not. Thus, given a compact non-positively curved space $X$,
we investigate the subgroups $\G\subset\pi_1X$
and ask
under what circumstances one can deduce that
$\G$ is the fundamental group of a
{\em compact} non-positively curved space.
For manifolds of dimension $\le 3$ and complexes of dimension
$\le 2$ it is shown that the finite presentability of a subgroup is
sufficient to guarantee
the existence of a cocompact action.
In higher dimensions one
encounters more subtle obstructions, for example higher finiteness
conditions, the complexity of decision problems in the subgroup,
and the structure of centralizers.
We prove some
general results concerning decision problems and use them
to investigate
certain closed
non-positively curved manifolds $M$ and subgroups $\G\subset
\pi_1M$ that arise as the fundamental groups
of closed aspherical submanifolds
$N_\G\subset M$; these
subgroups $\G$ enjoy properties reminiscent of
semihyperbolic groups
but they are not semihyperbolic.