On the equivariant K– and KO–homology of some special linear groups

Abstract

We compute the equivariant KO-homology of the classifying space for proper actions of SL 3 (Z) and GL 3 (Z). We also compute the Bredon homology and equivariant K-homology of the classifying spaces for proper actions of PSL 2 (Z[ 1 p ]) and SL 2 (Z[ 1 p ]) for each prime p. Finally, we prove the unstable Gromov-Lawson-Rosenberg conjecture for PSL 2 (Z[ 1 p ]) when p ≡ 11 (mod 12).

Cohomology of Fuchsian groups and non-Euclidean crystallographic groups

“…Other interpretations of the cohomology of Fuchsian groups have appeared in the literature. These have primarily dealt with lifting phenomena [17], with Eichler cohomology [5,6] or with K -theory in relation to the Baum-Connes conjecture [2,12,14].…”

Computations in $$C_{pq}$$ C pq -Bredon cohomology

“…The first computation of the Bredon chain and cochain complex was done in [58]. In detail, he determination of stabilizers, and conjugacy relations, as well as the differentials have been used as input for computations of Bredon cohomology with coefficients in complex representations in [58], for coefficients in twisted representations in [9], [10], and more recently, with the coefficients of equivariant real K-Theory in [31].…”