The bounded cohomology of the transformation groups of Euclidean spaces

In this Master thesis, we aim to study and discuss a result concerning the vanishing of the bounded cohomology of the transformation groups of Euclidean spaces. To this end, we first recall the definition of bounded cohomology, along with its main properties and key results. We then briefly review the notion of amenability and explore its connection with bounded cohomology. We then focus on the study of the bounded cohomology of transformation groups of Euclidean spaces. We begin with the group of homeomorphisms of R^n with compact support, as treated by Matsumoto and Morita in 1985, and later we focus our attention on the bounded cohomology of the full group of all the homeomorphisms (and diffeomorphisms) of Euclidean spaces. This latter result is recent and due Fournier-Facio, Monod, and Nariman (2024). Quite surprisingly, in both cases the bounded cohomology of the aforementioned transformation groups vanishes, although the techniques employed are substantially different.