VORONOI COMPLEXES IN HIGHER DIMENSIONS, COHOMOLOGY OF $GL_N(\mathbb{Z} )$ FOR $N\geqslant 8$ AND THE TRIVIALITY OF $K_8(\mathbb{Z} )$
Abstract We enumerate the low-dimensional cells in the Voronoi cell complexes attached to the modular groups $\mathit {SL}_N(\mathbb{Z} )$ and $\mathit {GL}_N(\mathbb{Z} )$ for $N=8,9,10,11$ , using quotient sublattice techniques for $N=8,9$ and linear programming methods for higher dimensions. These enumerations allow us to compute some cohomology of these groups and prove that $K_8(\mathbb{Z} ) = 0$ . We deduce from it new knowledge on the Kummer-Vandiver conjecture.
Authors
- Philippe Elbaz–Vincent (ORCID: https://orcid.org/0000-0002-8629-3021)
- Mathieu Dutour Sikirić (ORCID: https://orcid.org/0000-0001-7641-4785)
- Jacques Martinet
- Alexander Kupers (ORCID: https://orcid.org/0000-0003-0947-8820)
Institutions
- Université de Bordeaux (FR)
- University of Toronto (CA)
- Ruđer Bošković Institute (HR)
- Centre Émile Durkheim (FR)
- Institut des langues et cultures d'Europe, Amérique, Afrique, Asie et Australie (FR)
- Université Grenoble Alpes (FR)
Publication Details
- Journal
- Journal of the Institute of Mathematics of Jussieu
- Published
- 2025-11-24
- DOI
- https://doi.org/10.1017/s1474748025101394
- Primary Topic
- Advanced Algebra and Geometry
- Type
- article
- Field-Weighted Citation Impact
- 0.00