TITLE: Simplicial volume
QUESTION [9 upvotes]: Is there a finite dimensional closed manifold $M$ which is a $K(\pi,1)$, whose fundamental group is not word-hyperbolic, but which has a positive simplicial volume (ie "Gromov norm")?
(Added:) The answers of Jim and Richard are both excellent; another example is any closed, irreducible locally symmetric manifold (of non-positive curvature). But these examples are all CAT(0); I wonder if there is an example which is not CAT(0)? (of course then it is hard to see the example is a $K(\pi,1)$ . . .)

REPLY [2 votes]: Just take an irreducible, but not atoroidal, 3-manifold with the property that at least one of the pieces in the JSJ-decomposition is hyperbolic.
Such a manifold is nonpositively curved by Leeb's thesis, hence a $K(\pi,1)$. However the fundamental group is not hyperbolic because of the abelian subgroups coming from incompressible tori. Its simplicial volume is the sum of the simplicial volumina of the hyperbolic pieces, thus non-zero.