In this talk I will recall and discuss the conjecture of John Milnor on the homology of Lie groups made discrete, as well as its algebraic analogue, the Friedlander conjecture. In a work still partially in progress, we give a proof of that conjecture for algebraic groups G over algebraically closed fields. I will sketch some ideas behind this proof, in particular the role of A1-homotopy theory (already used by V. Voevodsky to prove other conjectures of John Milnor concerning mod 2 Galois cohomology and quadratic forms) and the role of a new object attached to G, its simplicial building. We will emphasize the case G = SL_2, SL_3,...