Geometry of Arithmetic Groups and Related Groups

本课题旨在研究两个关于大规模几何的基本问题：经典算算群SL(n，Z)， s -算算群SL（n,Z[1/p]）和函数域算算群SL(n,F[t])，其中F是一个有限域。第一个问题是确定等差群的Dehn函数和高维等周函数。第二是确定非紧函数-域算术群的上同调在由算术群的几何秩所给出的维数上是几乎无限产生的。PI还计划研究通用格的基本开放问题-例如SL(n,Z[t]) -以及关于有限性质和词度量的可解算术群。本文还计划对CAT(0)配合物上哪些晶格是有限生成的进行研究。矩阵是数字的数组。对控制矩阵行为的算术的研究——一项数学、物理科学和社会科学都从中受益匪浅的研究——是该奖项拟议研究的中心焦点。PI建议延续数学传统，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。

The PI intends to study two basic problems about the large-scale geometry of arithmetic groups: classical arithmetic groups such as SL(n,Z), S-arithmetic groups such as SL(n,Z[1/p]), and function-field-arithmetic groups such as SL(n,F[t]) where F is a finite field. The first problem is to identify the Dehn functions and higher dimensional isoperimetric functions for arithmetic groups. The second is to determine that the cohomology of non-cocompact function-field-arithmetic groups is virtually infinitely generated in the dimension given by the geometric rank of the arithmetic group. The PI also plans to investigate basic open questions for universal lattices - such as SL(n,Z[t]) - and solvable arithmetic groups concerning finiteness properties and word metrics. An investigation of which lattices on CAT(0) complexes are finitely generated is also planned.Matrices are arrays of numbers. The study of the arithmetic governing the behavior of matrices - a study that mathematics, the physical sciences, and the social sciences have benefited from greatly - is the central focus of the proposed research for this award. The PI proposes to continue the mathematical tradition of bundling the equations that represent the algebra of matrices into a single geometric theory, thus allowing techniques from geometry to deepen our understanding of algebra.