Simplicial Volume and Relative Bounded Cohomology

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.