Stability in Bounded Cohomology for Classical Groups, I: The Symplectic Case

Abstract

We show that continuous bounded group cohomology stabilizes along the sequences of real or complex symplectic Lie groups, and deduce that bounded group cohomology stabilizes along sequences of lattices in them, such as $(\mathrm{Sp}_{2r}(\mathbb{Z}))_{r \geq 1}$ or $(\mathrm{Sp}_{2r}(\mathbb{Z}[i]))_{r \geq 1}$. Our method is based on a general stability criterion which extends Quillen's method to the functional analytic setting of bounded cohomology. This criterion is then applied to a new family of complexes associated to symplectic polar spaces, which we call symplectic Stiefel complexes; similar complexes can also be defined for other families of classical groups acting on polar spaces.