Large-scale geometry of arithmetic groups and universal lattices

摘要：项目负责人：Kevin E. wortman，主要研究关于大规模几何的四个基本问题：“经典算算群”SL(n，Z)，“s算算群”asSL（n,Z[1/p]）和“函数-域算算群”asSL(n,F[t])，其中F是有限域。第一个问题是确定当n = 3时SL（n，Z）的Dehn函数。第二，确定给定的函数-域-算法群是FPm类型的哪一个m。第三，建立一个包含所有类型的算术群的填充理论，包括上面提到的第一个和第二个问题。第四，验证无限生成的函数-域-算术群的上同调的第一维。此外，PI打算利用从上述问题中学到的知识，进一步研究“泛格”(如SL（n,Z[t]）的有限性和词度量。上面所有的问题都包括对称为矩阵的数字数组的研究。能够理解矩阵的算术是数学中的一个基本问题，在大多数科学和经济学领域都有应用。正如数学家们在过去取得的巨大成功一样，pii打算通过将矩阵表示的方程捆绑成一个单一的几何理论来研究与矩阵相关的算术。

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.