We use bounded cohomology to define a notion of volume of an SO(n, 1)valued representation of a lattice Γ < SO(n, 1) and, using this tool, we give a complete proof of the volume rigidity theorem of Francaviglia and Klaff [19] in this setting. Our approach gives in particular a proof of Thurston's version of Gromov's proof of Mostow Rigidity (also in the non-cocompact case), which is dual to the Gromov-Thurston proof using the simplicial volume invariant.

This paper is devoted to the construction of norm-preserving maps between bounded cohomology groups. For a graph of groups with amenable edge groups we construct an isometric embedding of the direct sum of the bounded cohomology of the vertex groups in the bounded cohomology of the fundamental group of the graph of groups. With a similar technique we prove that if (X,Y) is a pair of CW-complexes and the fundamental group of each connected component of Y is amenable, the isomorphism between the relative bounded cohomology of (X,Y) and the bounded cohomology of X in degree at least 2 is isometric. As an application we provide easy and self-contained proofs of Gromov Equivalence Theorem and of the additivity of the simplicial volume with respect to gluings along \pi_1-injective boundary components with amenable fundamental group

If Γ < PSL(2, C) is a lattice, we define an invariant of a representation Γ → PSL(n, C) using the Borel class β (n) ∈ H 3 c (PSL(n, C), R). We show that this invariant satisfies a Milnor-Wod type inequality and its maximal value is attained precisely by the representations conjugate to the restriction to Γ of the irreducible complex n-dimensional representation of PSL(2, C) or its complex conjugate. Major ingredients of independent interest are the extension to degenerate configuration of flags of a cocycle defined by Goncharov and its study, as well as the identification of H 3 b (SL(n, C), R) as a normed space.

We define a notion of semi-conjugacy between orientationpreserving actions of a group on the circle, which for fixed point free actions coincides with a classical definition of Ghys. We then show that two circle actions are semi-conjugate if and only if they have the same bounded Euler class. This settles some existing confusion present in the literature.

We provide sharp lower bounds for the simplicial volume of compact 3‐manifolds in terms of the simplicial volume of their boundaries. As an application, we compute the simplicial volume of several classes of 3‐manifolds, including handlebodies and products of surfaces with the interval. Our results provide the first exact computation of the simplicial volume of a compact manifold whose boundary has positive simplicial volume. We also compute the minimal number of tetrahedra in a (loose) triangulation of the product of a surface with the interval.