Abstract: 
Which aspherical manifolds support metrics of non-positive
curvature? Which groups arise as the fundamental groups of such manifolds?
How does one go about deciding whether two "given" groups
or manifolds are "the same" (ie isomorphic in the appropriate category)?
And to what extent can such decision processes be simplified if one
restricts one's attention to spaces that support metrics of non-positive 
curvature?