Large-scale geometry of arithmetic groups and universal lattices

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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.