Small-dimensional classifying spaces for arithmetic subgroups of general linear groups (Q795180)

Let \(\Gamma\) be an arithmetic subgroup of the general linear group \(G({\mathbb{Q}})\) of a division algebra D of finite dimension over \({\mathbb{Q}}\), and let X be the symmetric space of maximal compact subgroups in the group \(G({\mathbb{R}})\) of real points of G. Generalizing a method of C. Soulé and J. Lannes, the author constructs a natural compact deformation retract Y of the quotient X/\(\Gamma\) such that the dimension of Y equals the virtual cohomological dimension of \(\Gamma\). The universal cover of Y has a structure as a cell-complex on which \(\Gamma\) acts cell-wise with finite stabilizers; this induces also a natural cell- complex structure on Y. This construction is of use in explicit computations of the cohomology of \(\Gamma\).