1996/11/19 by Avner Ash, Ash, Avner, Mark W. McConnell +1
Mathematics · #FOS: Mathematics #Representation Theory (math.RT) #math.RT
paper · pdf · doi:10.48550/arxiv.math/9611220
arxiv created 1996/11/19 · arxiv updated 2016/09/06
Let \bold G be a reductive algebraic group defined over \Q, and let Γ be an arithmetic subgroup of \bold G(\Q). Let X be the symmetric space for \bold G(\R), and assume X is contractible. Then the cohomology (mod torsion) of the space X/Γ is the same as the cohomology of Γ. In turn, X/Γ will have the same cohomology as W/Γ, if W is a ``spine'' in X. This means that W (if it exists) is a deformation retract of X by a Γ-equivariant deformation retraction, that W/Γ is compact, and that dim W equals the virtual cohomological dimension (vcd) of Γ. Then W can be given the structure of a cell complex on which Γ acts cellularly, and the cohomology of W/Γ can be found combinatorially.