Homology of Linear Groups with Applications to Algebraic K-theory

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Knudson studies the homology of linear groups over various rings of interest in algebraic K-theory. In this project he focuses attention on the low-dimensional homology groups of the general linear group over the coordinate ring of an elliptic curve over a number field with the goal of proving that the second K-group of such a curve has finite rank. This extends Knudson's previous work on the homology of rank one groups over such rings. In addition, a new homology theory based on algebraic cycles in the product of a scheme with the simplicial classifying scheme of an algebraic group is constructed and its properties investigated. The hope is that this construction will lead to a proof of the Friedlander-Milnor conjecture concerning the homology of algebraic groups made discrete. Also, the investigator's previous work on the structure of special linear groups over integral Laurent polynomial rings is used in the study of the Burau representation of the braid groups. This representation is known to be faithful for three strings and unfaithful for five or more strings; the remaining case of four strings is singled out for study in this project. Finally, Knudson studies the completion of a discrete group relative to a Zariski dense representation in a reductive group over a field of positive characteristic. This generalizes the classical unipotent completion of a group and extends to positive characteristic R. Hain's work in characteristic zero.A scheme is a geometric object constructed from solution sets of polynomial equations. Algebraic K-theory associates to a scheme a sequence of groups which encode information about the scheme. One aspect of this project is the study of the K-groups of an elliptic curve. Such curves have remarkably rich structure and appear in various branches of mathematics such as algebraic geometry and coding theory. A seemingly unrelated part of this project concerns the so-called Burau representation of the braid groups which is intimately connected with knot theory. These diverse topics are unified by studying the structure of groups of matrices with entries in various rings (the coordinate ring of the elliptic curve in the first case and the ring of integral Laurent polynomials in the second). The hope is to solve several outstanding conjectures about these objects.