Large-scale geometry of arithmetic groups and universal lattices

奖：DMS-0905891首席研究员：Kevin E.Wortman主要研究算术群的四个基本问题：“经典算术群”如SL(n，Z)、“S算术群”如ASSL(n，Z[1/p])和“函数域算术群”如ASSL(n，F[t])，其中F是有限域。第一个问题是确定SL(n，Z)的Dehn函数当n为3时。第二个问题是确定给定的函数域算术群对哪个m是FPM型的。第三，发展算术群的填充理论，它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，证明了无限生成的函数-域-算术群的上同调的第一维猜想。此外，PI还打算利用从上述问题中学到的知识来进一步研究SL(n，Z[t])等“泛格”的有限性质和字度量。上述所有问题都包括对称为矩阵的数组的研究。能够理解矩阵的算术是数学中的一个基本问题，在科学和经济的大多数领域都有应用。正如数学家在过去取得的巨大成功一样，PI打算通过将矩阵所代表的方程式捆绑到一个单一的几何理论中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.