Classical projective geometry and arithmetic groups
We construct cell complexes which are homotopic to certain locally symmetric spaces. Each cell complex is the geometric realization of a partially ordered set \({\mathcal C}\); the elements of \({\mathcal C}\) are configurations in projective space. The locally symmetric spaces we study are modular varieties \(K\setminus G/\Gamma\), with \(G=SL(n,{\mathbb{R}})\), \(n=2,3,4\), and \(\Gamma =\Gamma (p)\subset SL(n,{\mathbb{Z}})\) a principal congruence subgroup (p is an odd prime). Our cell complex is embedded in \(K\setminus G/\Gamma\) as a deformation retract. The deformation retract itself was found by Soulé and Lannes and studied by Ash. We generalize their work and include two main points of innovation. First, we have explicit results for the specific \(\Gamma\) we study. Second, we introduce the projective configurations in \({\mathcal C}\) as objects of study in their own right. As a by-product of this work we prove that the Soulé-Lannes retract is dual to the first Voronoi decomposition of \(K\setminus G\) for all n.
scientific article; zbMATH DE number 4210417
scientific article