Geometry and Cohomology of Arithmetic and Related Groups

University Of Utah, Salt Lake City UT

Investigators

Abstract

Matrices -- arrays of numbers -- are ubiquitous in science, engineering, and business. The study of the arithmetic governing the behavior of matrices has benefited all these application areas and has also led to advances in several branches of mathematics. This research project aims to advance work that is underway to bundle the equations that represent the algebra of matrices into a single geometric theory, thus allowing techniques from geometry to deepen our understanding of algebra. This research project aims to study the large-scale geometry, finiteness properties, and cohomological behavior of function-field-arithmetic groups such as the special linear groups SL(n,F[t]) where F is a finite field, and of related groups such as SL(n,Z[t]). Specifically, the principal investigator intends to investigate the cohomology with group ring coefficients of these families of groups. One would also like to understand when groups such as these special linear groups have projective resolutions or classifying complexes with finite skeleta through dimension m, and to determine coefficient modules M and natural numbers m for which the cohomology group of one of these special linear groups in degree m with coefficients in M is infinitely generated.