Abstract: Group cohomology allows one to associate to a group a sequence of algebraic invariants. A main tool to compute these invariants for the group $SL_n(\mathbb Z)$ are certain tessellations of the symmetric space associated to $SL_n(\mathbb R)$. I will indicate how these tessellations are defined and how they can be used to compute parts of the cohomology of $SL_n(\mathbb Z)$, both by hand and using computers.