• bounded cohomology of groups
• hyperbolic manifolds with geodesic boundary
• measure homology
• simplicial volume

The main aim of this thesis is the study of the (relative) bounded cohomology and of the simplicial volume of compact manifolds with boundary, with a particular attention devoted to hyperbolic manifolds with non-empty geodesic boundary.

Our approach to the computation of the simplicial volume of hyperbolic manifolds with non-empty geodesic boundary is grounded on a relative version of Thurston's smearing construction in the context of relative measure homology. Then we require to show that relative singular homology is isometric to relative measure homology, providing an extension of Loeh's result for a wide class of topological pairs.

As in the absolute case, we translate this issue in the dual setting of (continuous) bounded cochains. Therefore, it naturally arises the necessity to develop some new aspects of the theory of continuous bounded cohomology of topological pairs, which are probably of independent interest.
The understanding of the absolute case is mostly based on the relationship between bounded cohomology of spaces and of groups established by Gromov, Ivanov (and by Monod in the continuous case).
In the relative case, starting from a definition of bounded cohomology of pairs of groups due to Gromov, we provide an extension of Gromov's, Ivanov's and Monod's results for pairs of topological spaces.