Cohomology of $GL_N(\mathbb{Z})$ for $N>7$, trivialilty of $K_8(\mathbb{Z})$ and arithmetical applications

Using techniques from lattices and new methods for building the low-dimensional parts of the Voronoi complexes associated to  $GL_N(\mathbb{Z})$ for $N>7$, we show how we can get partial informations on the cohomology of $GL_N(\mathbb{Z})$
and use it to prove the triviality of $K_8(\mathbb{Z})$ and deduce some arithmetical consequences. We will also discuss the
algorithmic complexity of the methods involved.

This is  based on joint work with M. Dutour-Sikiric, S. Kupers and J. Martinet.