On Bounded-Cohomological Stability for Classical Groups

Abstract

Bounded cohomology, introduced independently and in different contexts by Johnson, Trauber, and Gromov in the late seventies and early eighties, is a very rich invariant of groups that can detect some of their coarse geometric features. Due to a lack of general computational tools, bounded cohomology remains in general obscure as of today. In the late nineties, Burger and Monod introduced the notion of continuous bounded cohomology, a suitable generalization for topological groups. Apart from having several interesting applications, this theory has remarkably shed light on the bounded cohomology of higher-rank lattices, which is determined to some extent by the continuous bounded cohomology of their ambient semisimple Lie groups. The relationship can be fully exploited in degree two: Being isomorphic to their second continuous cohomology, the second continuous bounded cohomology of any connected semisimple Lie group with finite center (or more generally, of any connected, simply connected, semisimple algebraic group over a local field) is completely understood. This fact and a few examples of groups in this class for which the isomorphism holds in degrees three or four, gave rise to the so-called isomorphism conjecture. Usually attributed to Dupont and Monod, it states that the isomorphism known in degree two should also hold in every degree. The ultimate goal of this thesis is to add further evidence to this conjectural picture: We prove the isomorphism conjecture in degree three for the family of complex symplectic groups. This will be obtained as corollary of the bounded-cohomological stability along said family. One says that continuous bounded cohomology is stable along an infinite nested sequence of topological groups if it is eventually constant in every degree. We develop here a machinery that gives bounded-cohomological stability along any sequence of locally compact, second-countable groups, provided that there exists a sequence of complexes on which the respective groups act. It is based on an original idea of Quillen in the setting of group homology, and on an ad hoc treatment in continuous bounded cohomology by Monod for the families of general and special linear groups. Our method improves Monod's stability range in degree three for special linear groups over non-Archimedean fields. Upon constructing a family of complexes that serves as an input to the aforementioned machinery, the so-called symplectic Stiefel complexes, we then prove stability of continuous bounded cohomology along the families of real and complex symplectic groups. While the stability range produced is insufficient to prove the isomorphism conjecture in degree three, we complete its proof in the complex case via a bootstrapping procedure. Based on a computation by Bucher-Burger-Iozzi, we moreover determine the Gromov norm of a generating class of the third continuous bounded cohomology. Inspired on the situation in continuous cohomology, continuous bounded cohomology is expected to be stable along all families of classical split groups over local fields (indexed by the rank). We conclude by explaining how our methods should extend to other families.