Stable representation theory of infinite discrete groups

The goal of this thesis is to study representations of infinite discrete groups from a homotopical viewpoint. Our main tool and object of study is Gunnar Carlsson's deformation K-theory, which provides a homotopy theoretical analogue of the classical representation ring. Deformation K-theory is a contravariant functor from discrete groups to connective O-spectra, and we begin by discussing a simple model for the zeroth space of this spectrum. We then investigate two related phenomena regarding deformation K -theory: Atiyah-Segal theorems and excision. Atiyah-Segal theorems, first studied by Sir Michael Atiyah and Graeme Segal for the classical representation ring, relate the deformation K-theory of a group to the complex K-theory of its classifying space. Excision relates the deformation K-theory of an amalgamation to the deformation K-theory of its factors.; We use Morse theory for the Yang-Mills functional (as developed by Atiyah, Raul Bott, Karen Uhlenbeck, Georgios Daskalopoulos and Johan Rade) to prove an Atiyah-Segal theorem for fundamental groups of compact, aspherical surfaces, and we prove that deformation K-theory is excisive on all free products. Combined with work of Tyler Lawson, the former result yields homotopical information about the stable coarse moduli space of surface-group representations.