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打算研究有关算术群的大规模几何形状的两个基本问题：经典的算术群，例如SL（N，Z），SL（N，Z [1/p]），例如SL（N，Z [1/P]），以及诸如SL（N，F [t]）之类的功能场（N，F [t]），其中F是有限的。第一个问题是确定算术组的DEHN函数和较高的等级等级函数。第二个是确定非共同函数函数 - 算术基团的共同体学实际上是在算术基团的几何级别给出的维度中无限生成的。 PI还计划研究通用晶格的基本开放问题，例如SL（N，Z [t]），以及有关有限属性和单词指标的可解决算术组。还计划了对CAT（0）络合物的晶格的调查，也计划有限地生成。对矩阵行为的算术进行的研究 - 一项研究，该研究是大大受益于该奖项的拟议研究的核心重点。 PI建议继续将代表矩阵代数的方程式捆绑到单个几何理论中的数学传统，从而允许从几何形状进行技术加深我们对代数的理解。