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Voronoi complexes in higher dimensions, cohomology of GL_N(Z) for N≥8 and the triviality of K₈(Z)
Mathieu Dutour Sikirić, Philippe Elbaz-Vincent, Alexander Kupers, Jacques Martinet
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We enumerate the low dimensional cells in the Voronoi cell complexes attached to the modular groups SL_N(Z) and GL_N(Z) for N=8,9,10,11, using quotient sublattices 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 K₈(Z) = 0. We deduce from it new knowledge on the Kummer-Vandiver conjecture.
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