Alexander D. Rahm

The Rigid Facets Subdivision Algorithm in GAP

Understanding the structure of the cohomology of arithmetic groups is a very important problem with relations to number theory and various K-theoretic areas. Explicit cohomology computations usually proceed via the study of the actions of the arithmetic groups on their associated symmetric spaces, and recent years have seen several advances in algorithmic computation of equivariant cell structures for these actions. To approach computations of Farrell-Tate and Bredon (co)homology of arithmetic groups, one needs cell complexes having a rigidity property: cell stabilizers must fix their cells pointwise. The previously known algorithms (using Voronoi decompositions and such techniques do not provide complexes with this rigidity property, and this leads to a significant bottleneck, both for the computation of Farrell-Tate cohomology (resp. the torsion at small prime numbers in group cohomology) of arithmetic groups as well as for the computation of Bredon homology. In theory, it is always possible to obtain this rigidity property via the barycentric subdivision. However, the barycentric subdivision of an n-dimensional cell complex can multiply the number of cells by (n+1)! and thus easily let the memory stack overflow. We present an algorithm implemented in GAP, called rigid facets subdivision, which subdivides cell complexes for arithmetic groups such that stabilisers fix their cells pointwise, but only leads to a controlled increase (in terms of sizes of stabilizer groups) in the number of cells, avoiding an explosion of the data volume. The GAP implementation of the algorithm shows that cases like PSL_4(Z) can effectively be treated with it, using commonly available machine resources.