My research is in topology and geometry—specifically, surgery theory and geometric group theory. I am particularly interested in group actions on manifolds.
There will be a pre-print of my thesis posted soon, but I will mention the main result here.
We say that a group G is Q-PD if it satisfies Poincare duality with rational coefficients (i.e., if its classifying space BG does). Examples include the fundamental groups of aspherical manifolds.
But there are other geometric examples: if a group G acts freely on a rationally-acyclic, rational homology manifold, then G is Q-PD. Does every Q-PD arise in this way—does every Q-PD group act on such an object?
The answer is no: lattices with torsion in semisimple Lie groups are counterexamples.