Homology of Linear Groups with Applications to Algebraic K-theory

Knudson在代数k理论中研究了各种感兴趣的环上的线性群的同调。在本项目中，他主要研究椭圆曲线坐标环上一般线性群的低维同调群，目的是证明椭圆曲线的第二个k群是有限秩的。这扩展了Knudson先前关于这类环上秩一群的同调性的工作。在此基础上，构造了一个新的基于代数环的同调理论，并研究了它的性质。希望这个构造能证明弗里德兰德-米尔诺关于离散代数群同调的猜想。此外，研究者之前关于积分Laurent多项式环上的特殊线性群的结构的工作也被用于辫群的Burau表示的研究。已知这种表示对于三个字符串是忠实的，对于五个或更多字符串是不忠实的；剩下的四个字符串在这个项目中被挑选出来进行研究。最后，Knudson研究了正特征域上约化群中离散群相对于Zariski稠密表示的补全。这推广了群的经典单能完备性，并将其推广到特征零点的正特征R. Hain的工作。方案是由多项式方程的解集构成的几何对象。代数k理论将一组组序列与一个方案联系起来，这些组序列对该方案的信息进行编码。本课题的一个方面是研究椭圆曲线的k群。这类曲线具有非常丰富的结构，出现在代数几何、编码理论等数学的各个分支中。这个项目的一个看似无关的部分涉及所谓的辫群的Burau表示，这与结理论密切相关。这些不同的主题通过研究在不同环（第一种情况是椭圆曲线的坐标环，第二种情况是积分劳伦多项式的环）中有条目的矩阵群的结构而统一起来。希望能解决关于这些天体的几个突出的猜想。

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.