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法的研究使所有这些应用领域受益，也导致了几个数学分支的进步。该研究项目旨在推进正在进行的工作，将表示矩阵代数的方程捆绑到一个单一的几何理论中，从而允许几何技术加深我们对代数的理解。本课题旨在研究函数-域算术群(如F为有限域的特殊线性群SL（n,F[t]）和相关群SL（n,Z[t]）的大尺度几何、有限性质和上同调行为。具体而言，主要研究者打算研究这些族的群环系数与群环系数的上同性。人们还想了解诸如这些特殊线性群之类的群何时具有射影分辨率或通过维数m对有限骨架的复合体进行分类，并确定系数模m和自然数m，对于这些系数为m的m次特殊线性群中的一个的上同调群是无限生成的。

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.