In this project we will study the algebraic K-theory of infinite groups Gamma with torsion. These K-groups have important topological applications in Wall's finiteness obstruction theory and hence in the problem of classifying high-dimensional manifolds of a fixed homotopy type. In this project we will use the Farrell-Jones isomorphism conjecture as a conceptual approach towards the computation of the algebraic K-theory of infinite groups with torsion. The Farrell-Jones conjecture states that the algebraic K-theory of the integral group ring of Gamma comes from the algebraic K-theory of the virtually cyclic subgroups of Gamma. This conjecture has been proven for large classes of groups, and leads to very concrete calculations. In this project we will use this conjecture to compute the lower algebraic K-groups of the following groups: hyperbolic n-simplex reflection groups, hyperbolic reflection groups, certain co-compact lattices in semi-simple real Lie groups of R-rank 2 and three-dimensional crystallographic groups (for all of these groups the conjecture is well known). We also intend to get some preliminary results on the K-theory of groups such as SL(4,Z), the mapping class group and three-dimensional orbifold groups (for these groups the Farrell-Jones Conjecture is still open). All of these groups are of great interest in low-dimensional topology.
Algebraic K-theory plays an important role in many areas of modern mathematics: most notably algebraic topology, number theory, and algebraic geometry. In topology, surgery theory is the name given to a collection of techniques used to produce one manifold from another in a controlled fashion. Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along the cut or boundary. With the development of this theory in the 1960's, many problems in high dimensional topology, in particular the classification of high dimensional manifolds, were found to have obstructions in (and were often classified by) the algebraic K-theory of the integral group ring of the fundamental group Gamma of the manifold being studied. As such, it is of great interest to obtain computations of these K groups for various groups Gamma.
