Geometry and Cohomology of Arithmetic and Related Groups

矩阵--数字数组--在科学、工程和商业中无处不在。对控制矩阵行为的算术的研究使所有这些应用领域都受益，也促进了数学的几个分支的进步。这个研究项目旨在推进正在进行的工作，将代表矩阵代数的方程式捆绑成一个单一的几何理论，从而使几何学的技术加深我们对代数的理解。这项研究的目的是研究函数域算术群的大尺度几何、有限性质和上同调行为，例如特殊的线性群SL(n，F[t])，其中F是有限域，以及相关群SL(n，Z[t])的上同调行为。具体地说，主要研究者打算研究这些群族的上同调与群环系数。人们还想了解诸如这些特殊线性群的群何时具有射影分解或通过维度m对有限骨架的复形进行分类，并确定系数模M和自然数m，对于这些系数模M和自然数m，这些m次特殊线性群之一的上同调群在M中的系数是无限生成的。

Matrices -- arrays of numbers -- are ubiquitous in science, engineering, and business. The study of the arithmetic governing the behavior of matrices has benefited all these application areas and has also led to advances in several branches of mathematics. This research project aims to advance work that is underway to bundle the equations that represent the algebra of matrices into a single geometric theory, thus allowing techniques from geometry to deepen our understanding of algebra. This research project aims to study the large-scale geometry, finiteness properties, and cohomological behavior of function-field-arithmetic groups such as the special linear groups SL(n,F[t]) where F is a finite field, and of related groups such as SL(n,Z[t]). Specifically, the principal investigator intends to investigate the cohomology with group ring coefficients of these families of groups. One would also like to understand when groups such as these special linear groups have projective resolutions or classifying complexes with finite skeleta through dimension m, and to determine coefficient modules M and natural numbers m for which the cohomology group of one of these special linear groups in degree m with coefficients in M is infinitely generated.