The Farrell--Tate and Bredon homology for PSL\_4(Z) via cell subdivisions

Abstract

We provide some new computations of Farrell--Tate and Bredon (co)homology for arithmetic groups. For calculations of Farrell--Tate or Bredon homology, one needs cell complexes wherecell stabilizers fix their cells pointwise. We provide two algorithms computing an efficient subdivision of a complex to achieve this rigidity property. Applying these algorithms to available cell complexes for PSL4(Z) provides computations of Farrell--Tate cohomology for small primes as well as the Bredon homology for the classifying spaces of proper actions with coefficients in the complex representation ring.