In this project we will continue studying the algebraic K-theory of infinite groups G 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. F. T. Farrell and L. Jones conjecture states that the algebraic K-theory of the integral group ring ZG comes from the algebraic K-theory of the virtually cyclic subgroups of G. This conjecture has been proven for large classes of groups, and leads to very concrete calculations. The proposed research will use this conjecture to compute the lower algebraic K-groups of the following groups among others: hyperbolic n-simplex reflection groups, hyperbolic reflection groups, three-dimensional and four-dimensional crystallographic groups, Bianchi groups, Hilbert modular groups, and three-dimensional orbifold groups. We also intend to get some preliminary results on the lower algebraic K-theory mapping class groups (for these groups the Farrell-Jones Conjecture is still open). All of these groups are of 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 G of the manifold being studied. As such, it is of great interest to obtain explicit computations of these K groups for various groups G.
