13 research outputs found
The mod 2 cohomology rings of SL\_2 of the imaginary quadratic integers
We establish general dimension formulae for the second page of the
equivariant spectral sequence of the action of the SL\_2 groups over imaginary
quadratic integers on their associated symmetric space. On the way, we extend
the torsion subcomplex reduction technique to cases where the kernel of the
group action is nontrivial. Using the equivariant and Lyndon-Hochschild-Serre
spectral sequences, we investigate the second page differentials and show how
to obtain the mod 2 cohomology rings of our groups from this information.Comment: Version post-print corrigeant de petits erreurs concernant
l'observation qui se r\'ef\`ere sur l'appendice, Journal of Pure and Applied
Algebra, Elsevier, 2015, Accepted for publication on July 21, 201
The corner poset with an application to an n-dimensional hypercube stacking puzzle
For any dimension n â„ 3, we establish the corner poset, a natural triangular poset structure on the corners of 2-color hypercubes. We use this poset to study a problem motivated by a classical cube stacking puzzle posed by Percy MacMahon as well as Eric Crossâs more recent âEight Blocks to Madness.â We say that a hypercube is 2-color when each of its facets has one of two colors. Given an arbitrary multiset of 2-color unit n-dimensional hypercubes, we investigate when it is possible to find a submultiset of 2n hypercubes that can be arranged into a larger hypercube of side length 2 with monochrome facets. Through a careful analysis of the poset and its properties, we construct interesting puzzles, find and enumerate solutions, and study the maximum size, S(n), for a puzzle that does not contain a solution. Further, we find bounds on S(n), showing that it grows as Î(n2n)
The mod 2 cohomology rings of congruence subgroups in the Bianchi groups
We provide new tools for the calculation of the torsion in the cohomology of
congruence subgroups in the Bianchi groups : An algorithm for finding
particularly useful fundamental domains, and an analysis of the equivariant
spectral sequence combined with torsion subcomplex
reduction.\_\_\_\_\_\_\_\_\_\_With an appendix by Bui Anh Tuan and Sebastian
Sch{\"o}nnenbec
The mod 2 cohomology rings of congruence subgroups in the Bianchi groups
We provide new tools for the calculation of the torsion in the cohomology of congruence subgroups in the Bianchi groups : An algorithm for finding particularly useful fundamental domains, and an analysis of the equivariant spectral sequence combined with torsion subcomplex reduction