The Degree-Three Bounded Cohomology of Complex Lie Groups of Classical Type

We establish Monod's isomorphism conjecture in degree-three bounded cohomology for every complex simple Lie group of classical type. Our main ingredient is a bounded-cohomological stability theorem with an optimal range in degree three that we bootstrap from previous stability results by the author and Hartnick. The bootstrapping procedure relies on the occurrence in our setting of a variant of the recently observed phenomenon of secondary stability in the sense of Galatius--Kupers--Randal-Williams.