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?



Dualizable classifying spaces
Any finitely presented group of type FL admits a finite classifying space. (A group $G$ is of type FL if $\mathbb{Z}$ admits a finite length resolution by finitely generated, free $\mathbb{Z}G$-modules.) This is Theorem VIII.7.1 in Brown's book "Cohomology of groups".


More examples are given in section VIII.9 of Brown. These include knot groups, finitely generated nilpotent groups, and certain arithmetic groups.


Also the non-orientable surfaces (other than $\mathbb{R}P^2$) are examples.


