Geometry of Arithmetic Groups and Related Groups

算术群及相关群的几何

• 批准号: 1206946
• 负责人: Kevin Wortman
• 金额: $ 18.5万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1206946&HistoricalAwards=false
• 关键词: Geometry; Arithmetic; Groups; Related

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.

PI打算研究关于算术群的大规模几何的两个基本问题：经典算术群如SL(n,Z)、S算术群如SL(n,Z[1/p])和函数域算术群如SL(n,F[t])，其中F是有限域。第一个问题是识别算术群的 Dehn 函数和高维等周函数。第二个是确定非余紧函数域算术群的上同调实际上是在算术群的几何秩给定的维数中无限生成的。 PI 还计划研究通用格的基本开放问题 - 例如 SL(n,Z[t]) - 以及有关有限性属性和字度量的可解算术群。还计划研究 CAT(0) 复合体上的哪些格是有限生成的。矩阵是数字数组。对控制矩阵行为的算术研究——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 建议继续将表示矩阵代数的方程捆绑到单一几何理论中的数学传统，从而使几何技术能够加深我们对代数的理解。