Large-scale geometry of arithmetic groups and universal lattices

算术群和通用格的大规模几何

• 批准号: 0905891
• 负责人: Kevin Wortman
• 金额: $ 14万
• 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: 2009
• 资助国家: 美国
• 起止时间: 2009-10-01 至 2012-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0905891&HistoricalAwards=false
• 关键词: Large; scale; geometry; arithmetic; groups

AbstractAward: DMS-0905891Principal Investigator: Kevin E. WortmanThe principal investigator intends to study four basic problemsabout the large-scale geometry of arithmetic groups: "classicalarithmetic groups" such as SL(n,Z), "S-arithmetic groups" such asSL(n,Z[1/p]), and "function-field-arithmetic groups" such asSL(n,F[t]) where F is a finite field. The first problem is toidentify the Dehn function for SL(n,Z) if n 3. Second, todetermine for which m a given function-field-arithmetic group isof type FPm. Third, to develop a theory of fillings forarithmetic groups that is inclusive of all types of arithmeticgroups, and that includes the first and second problems statedabove. Fourth, to verify what is conjectured to be the firstdimension of the cohomology of a function-field-arithmetic groupthat is infinitely generated. In addition, the PI intends to usethe knowledge learned from the problems above to furtherinvestigate the finiteness properties and word metrics of"universal lattices", such as SL(n,Z[t]).What all of the problems above include is the study of arrays ofnumbers, called matrices. Being able to understand the arithmeticof matrices is a fundamental problem in mathematics that hasapplications to most fields of science and economics. Asmathematicians have done in the past with great success, the PIintends to study the arithmetic associated with matrices bybundling the equations they represent into a single geometrictheory.

摘要获奖：DMS-0905891 首席研究员：Kevin E. Wortman 首席研究员打算研究算术群大规模几何的四个基本问题：“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”，例如 SL(n,F[t])，其中 F 是有限域。第一个问题是在 n 3 的情况下识别 SL(n,Z) 的 Dehn 函数。第二个问题是确定给定函数域算术群的 m 属于 FPm 类型。第三，发展一种算术群的填充理论，该理论包括所有类型的算术群，并且包括上述第一和第二个问题。第四，验证无限生成的函数域算术群的上同调的第一维猜想。此外，PI打算利用从上述问题中学到的知识来进一步研究“通用格”的有限性和词度量，例如SL(n,Z[t])。上述所有问题所包含的就是研究数字数组，称为矩阵。能够理解矩阵的算术是数学中的一个基本问题，它应用于科学和经济学的大多数领域。正如数学家过去取得的巨大成功一样，PI 打算通过将矩阵所表示的方程捆绑到一个几何理论中来研究与矩阵相关的算术。