Steinberg homology vanishes in a range for all reductive groups, and the double Tits building T^2(Z^n) is n-connected, refining the Church-Farb-Putman conjecture in degrees 1 and 2.
Voronoi complexes in higher dimensions, cohomology of $GL_N(Z)$ for $N\geq 8$ and the triviality of $K_8(Z)$
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abstract
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 $K_8(Z) = 0$. We deduce from it new knowledge on the Kummer-Vandiver conjecture.
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Homological vanishing for the Steinberg representation II: reductive groups and integral conjectures
Steinberg homology vanishes in a range for all reductive groups, and the double Tits building T^2(Z^n) is n-connected, refining the Church-Farb-Putman conjecture in degrees 1 and 2.