arXiv subjects
Publications and source records attributed to Francesco Milizia.
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.
On a negatively curved closed manifold, there exists a well-defined map $Ψ^\bullet$ associating to every closed differential form a bounded cohomology class via integration over straight simplices. Classes in the image of this map, which, a priori, depend on the fixed family of straight simplices, are usually called De Rham classes, and constitute an interesting subspace of bounded cohomology. In this paper we prove that, in sufficiently high degrees, $Ψ^\bullet$ is a homomorphism of algebras, i.e., it sends the wedge product of closed differential forms to the cup product of the associated bounded cohomology classes. The degree in which $Ψ^\bullet$ starts to preserve products depends on the boundedness of Jacobians of straight simplices. For the barycentric straightening introduced by Besson, Courtois and Gallot, this happens for degrees $\ge 3$. As a corollary, the cup product of two De Rham classes vanishes, provided that its degree exceeds the dimension of the manifold (and the degrees of both classes are $\geq 3$). This result complements vanishing results for the cup product of De Rham classes due to Marasco and to Battista et al.
For any geodesic metric space $X$, we give a complete cohomological characterisation of the hyperbolicity of $X$ in terms of vanishing of its second $\ell^{\infty}$-cohomology. We extend this result to the relative setting of $X$ with a collection of uniformly hyperbolic subgraphs. As an application, we give a cohomological characterisation of acylindrical hyperbolicity.
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 study the simplicial volume of manifolds obtained from Davis' reflection group trick, the goal being characterizing those having positive simplicial volume. In particular, we focus on checking whether manifolds in this class with nonzero Euler characteristic have positive simplicial volume (Gromov asked whether this holds in general for aspherical manifolds). This leads to a combinatorial problem about triangulations of spheres: we define a partial order on the set of triangulations -- the relation being the existence of a nonzero-degree simplicial map between two triangulations -- and the problem is to find the minimal elements of a specific subposet. We solve explicitly the case of triangulations of the two-dimensional sphere, and then perform an extensive analysis, with the help of computer searches, of the three-dimensional case. Moreover, we present a connection of this problem with the theory of graph minors.
We exhibit a finitely presented group whose second cohomology contains a weakly bounded, but not bounded, class. As an application, we disprove a long-standing conjecture of Gromov about bounded primitives of differential forms on universal covers of closed manifolds.
Given a group $G$ acting cocompactly on a smooth manifold $M$ by deck transformations, there is an integration map, defined recently by Kato, Kishimoto and Tsutaya, from the top-degree bounded de Rham cohomology of $M$ to the coinvariants $\ell^\infty(G)_G$. We generalize its definition and show that it is an isomorphism. In the presence of boundary, a relative version of bounded de Rham cohomology is considered.
We revisit Gersten's $\ell^\infty$-cohomology of groups and spaces, removing the finiteness assumptions required by the original definition while retaining its geometric nature. Mirroring the corresponding results in bounded cohomology, we provide a characterization of amenable groups using $\ell^\infty$-cohomology, and generalize Mineyev's characterization of hyperbolic groups via $\ell^\infty$-cohomology to the relative setting. We then describe how $\ell^\infty$-cohomology is related to isoperimetric inequalities. We also consider some algorithmic problems concerning $\ell^\infty$-cohomology and show that they are undecidable. In an appendix, we prove a version of the de Rham's theorem in the context of $\ell^\infty$-cohomology.