TITLE: Dualizable classifying spaces
QUESTION [7 upvotes]: If $G$ is a finitely generated free group, then its classifying space $B G$ can be presented as a finite CW complex (a finite bouquet of circles), and therefore is Spanier-Whitehead dualizable.  Are there any other discrete groups $G$ with the property that $B G$ is Spanier-Whitehead dualizable?

REPLY [2 votes]: This example is undoubtedly covered by John Klein's general response, but is somewhat illustrative in its own right.
By the Cartan-Hadamard theorem, the universal cover of every connected, non-positively curved manifold $M$ is homeomorphic to $\mathbb{R}^n$.  So if $\Gamma = \pi_1(M)$ is the fundamental group of $M$, then $M = \mathbb{R}^n / \Gamma = B \Gamma$ is the classifying space of $\Gamma$.  If $M$ is itself compact, it is dualisable.
One tends to consider the case when $\Gamma$ is a discrete subgroup of $Isom(\mathbb{H}^n)$, the group of isometries of hyperbolic space.  If $\Gamma$ acts freely and cocompactly on $\mathbb{H}^n$, then $B\Gamma$ is stably dualisable.