  This survey paper introduces to a technique called Torsion Subcomplex
Reduction (TSR) for computing torsion in the cohomology of discrete groups
acting on suitable cell complexes. TSR enables one to skip machine computations
on cell complexes, and to access directly the reduced torsion subcomplexes,
which yields results on the cohomology of matrix groups in terms of formulas.
TSR has already yielded general formulas for the cohomology of the tetrahedral
Coxeter groups as well as, at odd torsion, of SL2 groups over arbitrary number
rings. The latter formulas have allowed to refine the Quillen conjecture.
Furthermore, progress has been made to adapt TSR to Bredon homology
computations. In particular for the Bianchi groups, yielding their equivariant
K-homology, and, by the Baum-Connes assembly map, the K-theory of their reduced
C *-algebras. As a side application, TSR has allowed to provide dimension
formulas for the Chen-Ruan orbifold cohomology of the complexified Bianchi
orbifolds, and to prove Ruan's crepant resolution conjecture for all
complexified Bianchi orbifolds.
