Cohomology at infinity and the well-rounded retract for general Linear Groups

Abstract

Let $\bold G$ be a reductive algebraic group defined over $\Q$, and let $\Gamma$ 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 $X/\Gamma$ is the same as the cohomology of $\Gamma$. In turn, $X/\Gamma$ will have the same cohomology as $W/\Gamma$, if $W$ is a ``spine'' in $X$. This means that $W$ (if it exists) is a deformation retract of $X$ by a $\Gamma$-equivariant deformation retraction, that $W/\Gamma$ is compact, and that $\dim W$ equals the virtual cohomological dimension (vcd) of $\Gamma$. Then $W$ can be given the structure of a cell complex on which $\Gamma$ acts cellularly, and the cohomology of $W/\Gamma$ can be found combinatorially.