O. IntroductionStable range conditions on a ring R were devised by H. Bass in order to determine values of n for which every matrix in GL,(R) can be row reduced (by addition operations with coefficients from R) to a matrix with the same last row and column as the identity matrix 1,. In order to obtain analogous results for orthogonal groups, M.R. Stein defined "absolute stable range" conditions on a commutative ring R. Because he was working with group schemes, Stein did not consider absolute stable range conditions for noncommutative rings. Here we do so, and take up a corresponding stability question for orthogonal groups, namely cancellation of quadratic forms. For this we use a very general definition of quadratic form, which specializes to all classical examples.Sections 1, 2 and 3 contain definitions associated with, and computations of, absolute stable rank. Definitions associated with quadratic forms are introduced in Sections 4, 5, 6 and 7; and Section 8 is devoted to Witt cancellation.

This is an introduction to algebraic K-theory with no prerequisite beyond a first semester of algebra (including Galois theory and modules over a principal ideal domain). The presentation is almost entirely self-contained, and is divided into short sections with exercises to reinforce the ideas and suggest further lines of inquiry. No experience with analysis, geometry, number theory or topology is assumed. Within the context of linear algebra, K-theory organises and clarifies the relations among ideal class groups, group representations, quadratic forms, dimensions of a ring, determinants, quadratic reciprocity and Brauer groups of fields. By including introductions to standard algebra topics (tensor products, localisation, Jacobson radical, chain conditions, Dedekind domains, semi-simple rings, exterior algebras), the author makes algebraic K-theory accessible to first-year graduate students and other mathematically sophisticated readers. Even if your algebra is rusty, you can read this book; the necessary background is here, with proofs.

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For a finite volume geodesic polyhedron P in hyperbolic 3-space, with the property that all interior angles between incident faces are integral submultiples of Pi, there is a naturally associated Coxeter group generated by reflections in the faces. Furthermore, this Coxeter group is a lattice inside the isometry group of hyperbolic 3-space, with fundamental domain the original polyhedron P. In this paper, we provide a procedure for computing the lower algebraic K-theory of the integral group ring of such Coxeter lattices in terms of the geometry of the polyhedron P. As an ingredient in the computation, we explicitly calculate some of the lower K-groups of the dihedral groups and the product of dihedral groups with the cyclic group of order two.Comment: 35 pages, 2 figure

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