Homology of Linear Groups with Applications to Algebraic K-theory

Knudson在代数K-理论中研究各种感兴趣环上的线性群的同调。在这个项目中，他主要研究数域上椭圆曲线的坐标环上一般线性群的低维同调群，目的是证明这类曲线的第二个K-群有有限秩.这推广了Knudson以前关于这类环上秩一群的同调的工作。此外，基于代数群的单纯分类方案乘积的代数圈构造了一个新的同调理论，并研究了它的性质。希望这一构造将导致关于离散代数群的同调的Friedlander-Milnor猜想的证明。此外，作者在积分Laurent多项式环上的特殊线性群的结构方面的工作也被用于辫子群的Burau表示的研究。众所周知，这种表示对于三根弦是忠实的，对于五根或更多根弦是不忠诚的；剩下的四根弦的情况在这个项目中被挑出来研究。最后，Knudson研究了正特征域上约化群相对于Zariski稠密表示的离散群的完备性。这推广了经典的群的幂等完备性，并推广到正特征R.Hain在特征零点的工作。方案是由多项式方程的解集构造的几何对象。代数K-理论将编码关于方案的信息的群序列与方案相关联。这个项目的一个方面是研究椭圆曲线的K-群。这种曲线具有非常丰富的结构，出现在数学的各个分支中，如代数几何和编码理论。这个项目的一个看似无关的部分涉及到所谓的辫子群的Burau表示，它与纽结理论密切相关。这些不同的主题通过研究各种环(第一种情况下是椭圆曲线的坐标环，第二种情况是积分洛朗多项式环)中的元素的矩阵组的结构来统一。希望解决关于这些物体的几个悬而未决的猜想。

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.