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

## TL;DR

This paper proves that bounded cohomology stabilizes for sequences of symplectic Lie groups and their lattices, using a new stability criterion and symplectic complexes, advancing understanding of cohomological properties in classical groups.

## Contribution

Introduces a general stability criterion for bounded cohomology and applies it to symplectic groups using symplectic Stiefel complexes, extending Quillen's method.

## Key findings

- Bounded cohomology stabilizes along symplectic Lie groups.
- Stability extends to lattices like Sp_{2r}(Z) and Sp_{2r}(Z[i]).
- Develops symplectic Stiefel complexes for analysis.

## 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.