Abstract. Simplicial volumes measure the complexity of fundamental cycles of manifolds. In this article, we consider the relation between simplicial volume and two of its variants -the stable integral simplicial volume and the integral foliated simplicial volume. The definition of the latter depends on a choice of a measure preserving action of the fundamental group on a probability space.We show that integral foliated simplicial volume is monotone with respect to weak containment of measure preserving actions and yields upper bounds on (integral) homology growth.Using ergodic theory we prove that simplicial volume, integral foliated simplicial volume and stable integral simplicial volume coincide for closed hyperbolic 3-manifolds and closed aspherical manifolds with amenable residually finite fundamental group (being equal to zero in the latter case).However, we show that integral foliated simplicial volume and the classical simplicial volume do not coincide for hyperbolic manifolds of dimension at least 4.

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

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.

Let n 3, let M be an orientable complete finite-volume hyperbolic n-manifold with compact (possibly empty) geodesic boundary, and let Vol.M / and kM k be the Riemannian volume and the simplicial volume of M . A celebrated result by Gromov and Thurston states that if @M D ∅ then Vol.M /=kM k D v n , where v n is the volume of the regular ideal geodesic n-simplex in hyperbolic n-space. On the contrary, Jungreis and Kuessner proved that if @M ¤ ∅ then Vol.M /=kM k < v n .We prove here that for every Á > 0 there exists k > 0 (only depending on Á and n)As a consequence we show that for every Á > 0 there exists a compact orientable hyperbolic n-manifold M with nonempty geodesic boundary such that Vol.M /=kM k v n Á.Our argument also works in the case of empty boundary, thus providing a somewhat new proof of the proportionality principle for noncompact finite-volume hyperbolic n-manifolds without geodesic boundary. 53C23; 57N16, 57N65 Preliminaries and statementsLet X be a topological space, let Y Â X be a (possibly empty) subspace of X , and let R be a ring (in the present paper only the cases R D R and R D Z will be considered). For i 2 N we denote by C i .X I R/ the module of singular i -chains over R, ie the R-module freely generated by the set S i .X / of singular i -simplices with values in X . The natural inclusion of Y in X induces an inclusion of C i .Y I R/ into C i .X I R/, so it makes sense to define C i .X; Y I R/ as the quotient spaceThe homology of the resulting complex is the usual relative singular homology of the topological pair .X; Y / and will be denoted by H .X; Y I R/.

We study the construction of quasimorphisms on groups acting on trees introduced by Monod and Shalom, that we call median quasimorphisms, and in particular we fully characterise actions on trees that give rise to non-trivial median quasimorphisms. Roughly speaking, either the action is highly transitive on geodesics, it fixes a point in the boundary, or there exists an infinite family of non-trivial median quasimorphisms. In particular, in the last case the second bounded cohomology of the group is infinite dimensional as a vector space. As an application, we show that a cocompact lattice in a product of trees only has trivial quasimorphisms if and only if both closures of the projections on the two factors are locally ∞-transitive.