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We enumerate the low dimensional cells in the Voronoi cell complexes attachedto the modular groups $SL_N(Z)$ and $GL_N(Z)$ for $N=8,9,10,11$, using quotientsublattices techniques for $N=8,9$ and linear programming methods for higherdimensions. These enumerations allow us to compute some cohomology of thesegroups and prove that $K_8(Z) = 0$. We deduce from it new knowledge on theKummer-Vandiver conjecture.

Voronoi complexes in higher dimensions, cohomology of $GL_N(Z)$ for $N\geq 8$ and the triviality of $K_8(Z)$