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VORONOI COMPLEXES IN HIGHER DIMENSIONS, COHOMOLOGY OF
$GL_N(\mathbb{Z} )$ FOR 
$N\geqslant 8$ AND THE TRIVIALITY OF 
$K_8(\mathbb{Z} )$
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu , First View
- Published online by Cambridge University Press:
- 24 November 2025, pp. 1-26
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- Article
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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.
