Geometry of Arithmetic Groups and Related Groups

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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.