Weighted cohomology of arithmetic groups.

In this thesis the equivalence of two cohomology theories attached to an arithmetic group is shown and some applications are made. The first is the recently introduced weighted cohomology of Goresky, Harder, and MacPherson, defined via a complex of sheaves on the reductive Borel-Serre compactification of the locally symmetric space attached to the arithmetic group. It replaces the intersection cohomology of the Baily-Borel compactification when the locally symmetric space is not Hermitian. The second is weighted $L\sb2$ cohomology introduced in work of Franke on the computation of cohomology by automorphic forms. In the Hermitian case the former reduces (in a particular case) to the intersection cohomology of the Baily-Borel compactification and the latter reduces to the $L\sb2$ cohomology, recovering the Looijenga and Saper-Stern theorem (Zucker's conjecture). (This uses in a crucial way results of Looijenga and so is not a new proof of the conjecture.) The equivalence is proven by a local argument on the reductive Borel-Serre compactification which is a 'desingularization' (only topologically) of the Baily-Borel compactification. Both cohomology theories are interpreted via complexes of sheaves and then an explicit local quasiisomorphism is constructed between them using results of Borel and Franke and the geometry of the compactification. The equivalence is actually with a version of $L\sb2$ cohomology that is modified to be locally finite dimensional. Using this result the weighted cohomology groups, which are topological invariants, are interpreted as Lie algebra cohomology of spaces of automorphic forms for the arithmetic group. This interpretation is combined with spectral information on the modified $L\sb2$ spaces to make the connexion with usual $L\sb2$ cohomology. The first application is to Goresky and MacPherson's topological approach to Arthur's Lefschetz formula for Hecke operators. Another application is a formula for the local cohomology of the Baily-Borel compactification; this formula was established in special cases in different settings earlier. The methods are applied to study $L\sb{p}$ cohomology and similar identities with weighted cohomology groups are proven.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/105129/1/9635580.pdfDescription of 9635580.pdf : Restricted to UM users only