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The goal of the present paper is to push forward the frontiers ofcomputations on Farrell-Tate cohomology for arithmetic groups. The conjugacyclassification of cyclic subgroups is reduced to the classification of modulesof group rings over suitable rings of integers which are principal idealdomains, generalizing an old result of Reiner. As an example of thenumber-theoretic input required for the Farrell-Tate cohomology computations,we discuss in detail the homological torsion in PGL(3) over principal idealrings of quadratic integers, accompanied by machine computations in theimaginary quadratic case.

On Farrell-Tate cohomology of GL(3) over rings of quadratic integers