Weakly bounded cohomology classes and simplicial volume of Davis’ aspherical manifolds

In the first part of the thesis we study some "boundedness properties" of cohomology classes of groups and spaces. The main contribution we present in this respect is the first known example of a finitely presented group whose second cohomology contains a weakly bounded class which is not bounded. This example settles a question of Mikhail Gromov about bounded primitives of differential forms on universal covers of closed manifolds.In the second part we study the simplicial volume of aspherical manifolds obtained from Davis’ reflection group trick. In particular, we focus on checking whether manifolds in this class with nonzero Euler characteristic have nonzero simplicial volume, motivated by a famous question of Gromov. We approach the problem by introducing a partial order on the set of triangulations of spheres (a triangulated sphere is the input data needed by Davis' procedure to construct a manifold), the relation being the existence of a nonzero-degree simplicial map between two triangulations, and by studying a specific subposet by finding its minimal elements. We solve completely the case of triangulations of the two-dimensional sphere, and then perform an extensive analysis, also with the help of computer searches, of the three-dimensional case.