Bounded Cohomology of Pinched Negatively Curved 3-Manifolds with Positive Cheeger Constant

In our work, we study the bounded cohomology of a particular family of Riemannian manifolds. Bounded cohomology was first introduced in the late 1970s by Johnson and Trauber, as a tool to study the structure of Banach algebras. After the publication of the pioneering paper "\emph{Volume and Bounded Cohomology}" by Mikhail Gromov, who extended the definition of bounded cohomology to the realm of topological spaces and groups, several results were discovered that make bounded cohomology stand out from classical singular cohomology. By some classical results, it can be used to study the simplicial volume of manifolds, classify circle actions (Ghys), and for other applications. Following the path traced by Teruhiko Soma in the mid-to-late 1990s, we first focus on studying the bounded cohomology of low-dimensional hyperbolic manifolds. In particular, we show that under some tameness assumptions, the vanishing of the bounded fundamental class of a hyperbolic manifold $M$ of infinite volume is equivalent to its geometric finiteness. We then turn our attention towards a particular class of negatively curved manifolds, namely negatively pinched manifolds with infinite volume and positive Cheeger constant. The Cheeger constant $h(M)$ is a metric invariant first defined by Sullivan and Cheeger in the early 1970s, to study analytic and dynamical properties of maps on Riemannian manifolds; the positivity of $h(M)$ reflects certain tameness conditions on the ends of the manifolds, which in turn are related to the bounded exactness of the volume form, and the vanishing of the bounded fundamental class. While the equivalence of these three notions in the general case still remains conjectural, we present some results in this direction due to Kim Inkang and Kim Sungwoon (2015).