# Nikolaus conference 2017

Speaker: Alexander D. Rahm (Luxembourg)

Title: Torsion Techniques for Infinite Discrete Groups

Abstract:

This talk describes works involving a technique called Torsion Subcomplex Reduction (TSR), which was developed by the speaker 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 $$SL_2$$ groups over arbitrary number rings (in joint work of M. Wendt and the speaker). The latter formulas have allowed Wendt and the speaker 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, which would be very hard to compute directly. As a side application, TSR has allowed the speaker to provide dimension formulas for the Chen-Ruan orbifold cohomology of the complexified Bianchi orbifolds, and to prove (jointly with F. Perroni) Ruan's crepant resolution conjecture for all complexified Bianchi orbifolds.