• Gromov Proportionality Principle
• Simplicial Volume
• Relative Bounded Cohomology
• Bicombing

In the first part of this thesis we prove the Proportionality Principle for the Lipschitz simplicial volume without any restriction on curvature, thus generalizing the main result in a paper by Löh and Sauer. The cone procedure employed by Löh and Sauer - which is based on the uniqueness of geodesics in Hadamard manifolds - is replaced here by a local construction that exploits the local convexity of general Riemannian manifolds. Our approach restricts in particular to the closed case, thus giving a different proof of the classical Gromov Proportionality Principle. Some estimates of the Lipschitz simplicial volume for product of manifolds are also provided.

In the second part, we establish a bounded cohomology characterization of relative hyperbolicity for group pairs: A group pair (Γ,Γ′) is relatively hyperbolic iff the comparison map: H_b^k(Γ, Γ′; V) → H^k(Γ, Γ′; V) is surjective for any k ≥ 2 and a large class of coefficients V. The "only if" part of the theorem is stronger than the analogous one in a similar theorem by Mineyev and Yaman.