Resolutions for unit groups of orders

Abstract

We present a general algorithm for constructing a free resolution for unit groups of orders in semisimple rational algebras. The approach is based on computing a contractible $G$-complex employing the theory of minimal classes of quadratic forms and Opgenorth's theory of dual cones. The information from the complex is then used together with Wall's perturbation lemma to obtain the resolution

Introductory computations in the cohomology of arithmetic groups

“…Theorems 12.1, 12.2 can be extended to the case where Z is replaced by O d and implemented on a computer as a method for determining the contractible CW-complex T . A good account of this approach is given in [18,36,6,26]. Using this approach, Sebastian Schönnenbeck has computed a complex T for various groups SL 2 (O d ) and stored its details as part of a library in HAP.…”

The mod 2 cohomology rings of congruence subgroups in the Bianchi groups

“…We use Sebastian Schönnenbeck's implementation [21] of the Voronoï cell complex to compute the cohomology of a sample (six levels η in each of ten imaginary quadratic fields) of congruence subgroups Γ 0 (η). Then we use Bui Anh Tuan's implementation of Rigid Facets Subdivision in order to extract the non-central 2torsion subcomplex.…”

“…• Grunewald's method of taking a presentation for the whole Bianchi group, and deriving presentations for finite index subgroups via the Reidemeister-Schreier algorithm [8]; • Utilizing the Eckmann-Shapiro lemma for computing cohomology of congruence subgroups directly from cohomological data of the full Bianchi group [20]; • Construction of a Voronoï cell complex for the congruence subgroup [4,21]. What one typically harvests with these approaches are tables of machine results in which everything looks somewhat ad hoc.…”

The Farrell–Tate and Bredon homology for PSL4(Z) via cell subdivisions

“…We would like to thank Graham Ellis for having supported the development of the first implementation of our algorithms -the "Torsion Subcomplexes Subpackage" for his Homological Algebra Programming (HAP) package in GAP. Further thanks go to Sebastian Schönnenbeck [1,19] for having provided us cell complexes for congruence subgroups in SL 3 (Z), used for benchmarking purposes in this paper, and especially to Mathieu Dutour Sikirić for having provided the cell complexes for SL 3 (Z), Sp 4 (Z) and PSL 4 (Z) in HAP. This article is dedicated to the memory of Aled Ellis.…”