arXiv subjects

Roberto Frigerio

Publications and source records attributed to Roberto Frigerio.

At least 19 recordsLinked to original sources

arXiv2026

The bounded area class of negatively curved surfaces

Let $S$ be an oriented surface, possibly of infinite type, endowed with a complete Riemannian metric with pinched negative curvature. We prove that the area form defines a non-trivial class in the second bounded cohomology group of $S$, unless $S$ is diffeomorphic to the disc or the cylinder. This is in sharp contrast with the $n$-dimensional case, $n>2$, where the non-triviality of the volume form in bounded cohomology is related to the Cheeger constant of the manifold. We also discuss how the bounded area class depends on the metric: we prove that, for compact surfaces, it recognizes constant curvature metrics among pinched negatively curved ones, while, even in the case of surfaces of infinite type, it does not distinguish non-isometric hyperbolic structures. More precisely, when $S$ is compact we show that, among the negatively curved structures of fixed area, the ones with constant curvature provide the unique minimizers for the norm of the bounded area class.

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arXiv2026

On the cup product of De Rham classes in bounded cohomology

On a negatively curved closed manifold, there exists a well-defined map $\Psi^\bullet$ 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, $\Psi^\bullet$ 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 $\Psi^\bullet$ 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.

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arXiv2026

Bounded cohomology classes from differential forms

Let $M$ be a complete hyperbolic $n$-manifold, $n\geq 2$. Via integration over geodesic simplices, any closed bounded differential 2-form on $M$ defines a bounded cohomology class in $H^2_b(M)$. It was proved by Barge and Ghys (for $n=2$) and by Battista et al. (for $n>2$) that, if $M$ is closed, then this procedure defines an injective embedding of the (infinite-dimensional) space of closed differential $2$-forms on $M$ into $H^2_b(M)$. We extend this result to the case when the fundamental group of $M$ is of the first kind, i.e. its limit set is equal to the whole boundary at infinity of hyperbolic space (this holds, for example, when $M$ has finite volume). Our argument is different from Barge and Ghys' original one, and relies on the following fact of independent interest: an $L^\infty$ function on the hyperbolic plane is determined by its integrals over all ideal triangles. We prove this fact by way of Fourier analysis on the hyperbolic plane.

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arXiv2024

Integral filling volume, complexity and integral simplicial volume of 3-dimensional mapping tori

We show that the integral filling volume of a Dehn twist $f$ on a closed oriented surface vanishes, i.e. that the integral simplicial volume of the mapping torus with monodromy $f^n$ grows sublinearly with respect to $n$. We deduce a complete characterization of mapping classes on surfaces with vanishing integral filling volume and, building on results by Purcell and Lackenby on the complexity of mapping tori, we show that, in dimension three, complexity and integral simplicial volume are not Lipschitz equivalent.

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arXiv2023

Efficient cycles of hyperbolic manifolds

Let $N$ be a complete finite-volume hyperbolic $n$-manifold. An efficient cycle for $N$ is the limit (in an appropriate measure space) of a sequence of fundamental cycles whose $\ell^1$-norm converges to the simplicial volume of $N$. Gromov and Thurston's smearing construction exhibits an explicit efficient cycle, and Jungreis and Kuessner proved that, in dimension $n\geq 3$, such cycle actually is the unique efficient cycle for a huge class of finite volume hyperbolic manifolds, including all the closed ones. In this paper we prove that, for $n\geq 3$, the class of finite-volume hyperbolic manifolds for which the uniqueness of the efficient cycle does not hold is exactly the commensurability class of the figure-8 knot complement (or, equivalently, of the Gieseking manifold).

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arXiv2022

Length functions on mapping class groups and simplicial volumes of mapping tori

Let $M$ be a closed orientable manifold. We introduce two numerical invariants, called filling volumes, on the mapping class group $\mathrm{MCG}(M)$ of $M$, which are defined in terms of filling norms on the space of singular boundaries on $M$, both with real and with integral coefficients. We show that filling volumes are length functions on $\mathrm{MCG}(M)$, we prove that the real filling volume of a mapping class $f$ is equal to the simplicial volume of the corresponding mapping torus $E_f$, while the integral filling volume of $f$ is not smaller than the stable integral simplicial volume of $E_f$. We discuss several vanishing and non-vanishing results for the filling volumes. As applications, we show that the hyperbolic volume of $3$-dimensional mapping tori is not subadditive with respect to their monodromy, and that the real and the integral filling norms on integral boundaries are often non-biLipschitz equivalent.

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arXiv2022

Central extensions and bounded cohomology

It was shown by Gersten that a central extension of a finitely generated group is quasi-isometrically trivial provided that its Euler class is bounded. We say that a finitely generated group $G$ satisfies Property QITB (quasi-isometrically trivial implies bounded) if the Euler class of any quasi-isometrically trivial central extension of $G$ is bounded. We exhibit a finitely generated group $G$ which does not satisfy Property QITB. This answers a question by Neumann and Reeves, and provides partial answers to related questions by Wienhard and Blank. We also prove that Property QITB holds for a large class of groups, including amenable groups, right-angled Artin groups, relatively hyperbolic groups with amenable peripheral subgroups, and 3-manifold groups. Finally, we show that Property QITB holds for every finitely presented group if a conjecture by Gromov on bounded primitives of differential forms holds as well.

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arXiv2021

The simplicial volume of contractible 3-manifolds

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.

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arXiv2019

Gromov's theory of multicomplexes with applications to bounded cohomology and simplicial volume

The simplicial volume is a homotopy invariant of manifolds introduced by Gromov in 1982. In order to study its main properties, Gromov himself initiated the dual theory of bounded cohomology, that developed into an active and independent research field. Gromov's theory of bounded cohomology was based on the use of multicomplexes, which are simplicial structures that generalize simplicial complexes without allowing all the degeneracies appearing in simplicial sets. In the first part of this paper we lay the foundation of the theory of multicomplexes. We construct the singular multicomplex K(X) associated to a topological space X, and we prove that K(X) is homotopy equivalent to $X$ for every CW complex X. Following Gromov, we introduce the notion of completeness, which translates into the context of multicomplexes the Kan condition for simplicial sets. We then develop the homotopy theory of complete multicomplexes. In the second part we apply the theory of multicomplexes to the study of the bounded cohomology of topological spaces. We provide complete proofs of Gromov's Mapping Theorem (which implies that the bounded cohomology of a space only depends on its fundamental group) and of Gromov's Vanishing Theorem, which ensures the vanishing of the simplicial volume of closed manifolds admitting an amenable cover of small multiplicity. The third part is devoted to the study of locally finite chains on non-compact spaces. We expand some ideas of Gromov to provide complete proofs of a criterion for the vanishing and a criterion for the finiteness of the simplicial volume of open manifolds. As a by-product of these results, we prove a criterion for the l^1-invisibility of closed manifolds in terms of amenable covers. As an application, we give the first complete proof of the vanishing of the simplicial volume of the product of three open manifolds.

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arXiv2019

A remark on the Mayer-Vietoris double complex for singular cohomology

Given an open cover of a paracompact topological space X, there are two natural ways to construct a map from the cohomology of the nerve of the cover to the cohomology of X. One of them is based on a partition of unity, and is more topological in nature, while the other one relies on the Mayer-Vietoris double complex, and has a more algebraic flavour. In this paper we prove that these two maps coincide, thus answering a question posed by N. V. Ivanov.

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arXiv2019

Amenable covers and l^1-invisibility

Let $X$ be a topological space admitting an amenable cover of multiplicity $k\in\mathbb{N}$. We show that, for every $n\geq k$ and every $α\in H_n(X;\mathbb{R})$, the image of $α$ in the $\ell^1$-homology module $H_n^{\ell^1}(X;\mathbb{R})$ vanishes. This strenghtens previous results by Gromov and Ivanov, who proved, under the same assumptions, that the $\ell^1$-seminorm of $α$ vanishes.

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arXiv2019

On volumes of hyperideal tetrahedra with constrained edge lengths

Hyperideal tetrahedra are the fundamental building blocks of hyperbolic 3-manifolds with geodesic boundary. The study of their geometric properties (in particular, of their volume) has applications also in other areas of low-dimensional topology, like the computation of quantum invariants of 3-manifolds and the use of variational methods in the study of circle packings on surfaces. The Schläfli formula neatly describes the behaviour of the volume of hyperideal tetrahedra with respect to dihedral angles, while the dependence of volume on edge lengths is worse understood. In this paper we prove that, for every $\ell<\ell_0$, where $\ell_0$ is an explicit constant, regular hyperideal tetrahedra of edge length $\ell$ maximize the volume among hyperideal tetrahedra whose edge lengths are all not smaller than $\ell$. This result provides a fundamental step in the computation of the ideal simplicial volume of an infinite family of hyperbolic 3-manifolds with geodesic boundary.

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arXiv2019

Ideal simplicial volume of manifolds with boundary

We define the ideal simplicial volume for compact manifolds with boundary. Roughly speaking, the ideal simplicial volume of a manifold $M$ measures the minimal size of possibly ideal triangulations of $M$ "with real coefficients", thus providing a variation of the ordinary simplicial volume defined by Gromov in 1982, the main difference being that ideal simplices are now allowed to appear in representatives of the fundamental class. We show that the ideal simplicial volume is bounded above by the ordinary simplicial volume, and that it vanishes if and only if the ordinary simplicial volume does. We show that, for manifolds with amenable boundary, the ideal simplicial volume coincides with the classical one, whereas for hyperbolic manifolds with geodesic boundary it can be strictly smaller. We compute the ideal simplicial volume of an infinite family of hyperbolic $3$-manifolds with geodesic boundary, for which the exact value of the classical simplicial volume is not known, and we exhibit examples where the ideal simplicial volume provides shaper bounds on mapping degrees than the classical simplicial volume.

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arXiv2017

The zero norm subspace of bounded cohomology of acylindrically hyperbolic groups

We construct combinatorial volume forms of hyperbolic three manifolds fibering over the circle. These forms define non-trivial classes in bounded cohomology. After introducing a new seminorm on exact bounded cohomology, we use these combinatorial classes to show that, in degree 3, the zero norm subspace of the bounded cohomology of an acylindrically hyperbolic group is infinite dimensional. In the appendix we use the same techniques to give a cohomological proof of a lower bound, originally due to Brock, on the volume of the mapping torus of a cobounded pseudo-Anosov homeomorphism of a closed surface in terms of its Teichmüller translation distance.

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arXiv2016

Bounded cohomology of discrete groups

Bounded cohomology of groups was first defined by Johnson and Trauber during the seventies in the context of Banach algebras. As an independent and very active research field, however, bounded cohomology started to develop in 1982, thanks to the pioneering paper "Volume and Bounded Cohomology" by M. Gromov, where the definition of bounded cohomology was extended to deal also with topological spaces. The aim of this monograph is to provide an introduction to bounded cohomology of discrete groups and of topological spaces. We also describe some applications of the theory to related active research fields (that have been chosen according to the taste and the knowledge of the author). The book is essentially self-contained. Even if a few statements do not appear elsewhere and some proofs are slighlty different from the ones already available in the literature, the monograph does not contain original results. In the first part of the book we settle the fundamental definitions of the theory, and we prove some (by now classical) results on low-dimensional bounded cohomology and on bounded cohomology of topological spaces. Then we describe how bounded cohomology has proved useful in the study of the simplicial volume of manifolds, for the classification of circle actions, for the definition and the description of maximal representations of surface groups, and in the study of higher rank flat vector bundles (also in relation with the Chern conjecture).

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arXiv2016

A note on semi-conjugacy for circle actions

We define a notion of semi-conjugacy between orientation-preserving 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.

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arXiv2016

Quasi-isometric rigidity of piecewise geometric manifolds

Two groups are virtually isomorphic if they can be obtained one from the other via a finite number of steps, where each step consists in taking a finite extension or a finite index subgroup (or viceversa). Virtually isomorphic groups are always quasi-isometric, and a group G is quasi-isometrically rigid if every group quasi-isometric to G is virtually isomorphic to G. In this survey we describe quasi-isometric rigidity results for fundamental groups of manifolds which can be decomposed into geometric pieces. After stating by now classical results on lattices in semisimple Lie groups, we focus on the class of fundamental groups of $3$-manifolds, and describe the behaviour of quasi-isometries with respect to the Milnor-Kneser prime decomposition (following Papasoglu and Whyte) and with respect to the JSJ decomposition (following Kapovich and Leeb). We also discuss quasi-isometric rigidity results for fundamental groups of higher dimensional graph manifolds, that were recently defined by Lafont, Sisto and the author. Our main tools are the study of geometric group actions and quasi-actions on Riemannian manifolds and on trees of spaces, via the analysis of the induced actions on asymptotic cones.

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arXiv2015

Integral Foliated Simplicial Volume of Aspherical Manifolds

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.

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