The PI proposes to study the deformation K-theory of a discrete group and relate this to the topological K-theory of its classifying space. This is analogous to the Atiyah-Segal theorem for compact Lie groups, with deformation K-theory playing the role of the complex representation ring. The PI believes that these calculations will shed light on the topology of the stable moduli space of flat connections for surfaces as well as Casson invariants of three manifolds.
Representations of groups have been important throughout much of mathematics as well as in physics and other sciences. The PI proposes to extend our understanding from finite and compact Lie groups to infinite discrete groups, using a variant of earlier approaches which relate representations to topological invariants. These results would add to our knowledge of surfaces as well as manifolds of dimensions three.
