Vanishing for the cohomology of hyperbolic 7-manifolds coming from triality

This is common work with Nicolas Bergeron. The last hyperbolic compact manifolds for which
Thurston's conjecture, on the existence of covers with non-trivial $H^1$, is unknown are the manifolds
of dimension 7 associated to triality orthogonal groups. I will sketch the proof of the negative answer
when congruence groups are considered. This result was announced by us some time ago, but the proof
had to wait for the general case of the twisted trace formula, now established by Moeglin and Waldspurger.