Homology of classical groups and K-theory

Abstract

The study of the homology groups of classical group over a ring R with coefficient A, where A is a commutative ring with trivial group action, seems important, notably because of their close relation to algebraic and Hermitian Ktheory and their appearance in the study of scissors congruence of polyhedra. Unfortunately, these groups are much too big and complicated to be computed explicitly. Therefore all results allowing to compare the ith homology groups of the same type of groups but with different size become quite important. In this thesis we mostly study the homology stability problem for the unitary groups, for example symplectic groups. In the first main theorem we prove that homology stabilizes for the hyperbolic unitary groups over rings with finite unitary stable rank, for example over rings with finite Krull dimension. In the second main theorem with assuming some additional assumptions on the ring we get a better range of homology stability. Homology stability type theorems have many important and interesting applications which the reader can find some of these applications in this thesis.