arXiv subjects
Publications and source records attributed to Giuseppe Bargagnati.
We prove that the minimal volume entropy of mapping tori over oriented closed smooth $3$-manifolds vanishes. Our approach uses a variation of the amenable category and a suitable version of the minimal volume entropy of a homology class introduced by Babenko and Sabourau.
We prove the vanishing of bounded cohomology of the groups acting on trees with almost prescribed local actions $G(F, F')$, where $F<F'$ are finite permutation groups such that $F'$ is 2-transitive. By contrast, when $F'$ is not 2-transitive, we prove that the second bounded cohomology with real coefficients of the groups $G(F, F')$ is infinite dimensional.
We study the action of the mapping class group on the subspace of de Rham classes in the degree-two bounded cohomology of a hyperbolic surface. In particular, we show that the only fixed nontrivial finite-dimensional subspace is the one generated by the Euler class. As a consequence, we get that the action of the mapping class group on the space of de Rham quasimorphisms has no fixed points.
We show that for $n \neq 1,4$ the simplicial volume of an inward tame triangulable open $n$-manifold $M$ with amenable fundamental group at infinity at each end is finite; moreover, we show that if also $π_1(M)$ is amenable, then the simplicial volume of $M$ vanishes. We show that the same result holds for finitely-many-ended triangulable manifolds which are simply connected at infinity.
We show that the simplicial volume of a contractible 3-manifold not homeomorphic to $\mathbb{R}^3$ is infinite. As a consequence, the Euclidean space may be characterized as the unique contractible $3$-manifold with vanishing minimal volume, or as the unique contractible $3$-manifold supporting a complete finite-volume Riemannian metric with Ricci curvature uniformly bounded from below. On the contrary, we show that in every dimension $n\geq 4$ there exists a contractible $n$-manifold with vanishing simplicial volume not homeomorphic to $\mathbb{R}^n$. We also compute the spectrum of the simplicial volume of irreducible open 3-manifolds.