Geometry of Arithmetic Groups and Related Groups

PI旨在研究大规模算术群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])，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。第二个是确定非上紧函数域算术群的上同调在算术群的几何秩所给出的维上无限生成。PI还计划研究泛格的基本开放问题--如SL(n，Z[t])--以及与有限性质和字度量有关的可解算术群。还计划研究CAT(0)复形上的哪些格是有限生成的。对支配矩阵行为的算术的研究-这项研究使数学、物理科学和社会科学受益匪浅-是该奖项拟议研究的中心重点。PI建议继续将代表矩阵代数的方程式捆绑成单一几何理论的数学传统，从而允许几何学的技术加深我们对代数的理解。

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.