Homology of Linear Groups

1. Topological Methods.- 1.1. Finite Fields.- 1.2. Quillen's Conjecture.- 1.3. Étale homotopy theory.- 1.4. Analytical Methods.- 1.5. Unstable Calculations.- 1.6. Congruence Subgroups.- Exercises.- 2. Stability.- 2.1. van der Kallen's Theorem.- 2.2. Stability for rings with many units.- 2.3. Local rings and Milnor K-theory.- 2.4. Auxiliary stability results.- 2.5. Stability via Homotopy.- 2.6. The Rank Conjecture.- Exercises.- 3. Low-dimensional Results.- 3.1. Scissors Congruence.- 3.2. The Bloch Group.- 3.3. Extensions and Generalizations.- 3.4. Invariants of hyperbolic manifolds.- Exercises.- 4. Rank One Groups.- 4.1. SL2(?[1/p]).- 4.2. The Bruhat-Tits Tree.- 4.3. SL2(k[t]).- 4.4. SL2(k[t, t?1]).- 4.5. Curves of Higher Genus.- 4.6. Groups of Higher Rank.- Exercises.- 5. The Friedlander-Milnor Conjecture.- 5.1. Lie Groups.- 5.2. Groups over Algebraically Closed Fields.- 5.3. Rigidity.- 5.4. Stable Results.- 5.5. H1, H2, and H3.- Exercises.- Appendix A. Homology of Discrete Groups.- A.1. Basic Concepts.- A.2. Spectral Sequences.- B.1. Classifying Spaces.- Appendix C. Étale Cohomology.- C.1. Étale Morphisms and Henselian Rings.- C.2. Étale Cohomology.- C.3. Simplicial Schemes.

"A book for graduates and researchers in K-theory, cohomology, algebraic geometry and topology. The theme is the development of the computing of the homology of the groups of matrices from Daniel Quillen's definitions of the higher algebraic K-groups. Stability theorems, low-dimensional results and the Friedlander-Milnor conjecture are discussed in this monograph."

-Aslib Book Guide

"This marks the first time that many of these results have been collected in a single volume..."

-Mathematical Reviews