arXiv subjects

Nicola Cavallucci

Publications and source records attributed to Nicola Cavallucci.

At least 19 recordsLinked to original sources

arXiv2026

The entropy of Gromov-Thurston manifolds and branched coverings

We develop a general theory for the dynamics of the geodesic flow of locally CAT(k) branched coverings and we show that it does not depend on the specific covering but only on the base space, the branching set and the degree of the covering. We find an explicit formula for the entropy: it equals the topological pressure of the associated natural dynamical system on the space of broken geodesics with a natural geometric potential. We find the exact asymptotic of the entropy as the number of sheets of the branched covering goes to infinity, as well as asymptotic properties of the measures of maximal entropy.

math.DS↗

arXiv2026

Planar lamplighter is not of negative type

The lamplighter group over the planar integer grid is proved to not be bi-Lipschitz equivalent to any metric space of negative type, so in particular it does not admit a bi-Lipschitz embedding into $L_1$. This shows the existence of finitely generated metabelian groups on which word metrics are never comparable up to constant factors to conditionally negative definite (CND) kernels, and that the property of admitting a word metric-comparable CND kernel is not preserved by wreath products.

math.MG↗

arXiv2026

CAT(0) spaces quasi-isometric to Euclidean spaces

We show that if a proper, geodesically complete, CAT(0) homology manifold is quasi-isometric to the Euclidean space R^n then it is homeomorphic to R^n. On the other hand, we show that there exist proper, geodesically complete, CAT(0) spaces quasi-isometric to R^n, which are not homeomorphic to it. We prove that our example is sharp in a suitable sense. Finally, we provide an example of a sequence of proper, geodesically complete, CAT(0) spaces that are not homology manifolds and that converge in the Gromov-Hausdorff sense to a topological manifold: this shows that the set of topological manifolds is not open in the class of proper, geodesically complete, CAT(0) spaces.

math.MG↗

arXiv2026

Sobolev-to-Lipschitz property of geodesically complete spaces with curvature bounded from above

We prove that every length space with curvature bounded from above that is geodesically complete has the Sobolev-to-Lipschitz property with exponent infinity. That is, every Sobolev map in the $W^{1,\infty}$-space has a Lipschitz representative so that the Lipschitz constant coincides with the infinity energy of the map. The proof is geometric and relies on arbitrarily small perturbations of geodesics to a curve that has zero length on the singular set. The motivation is to develop the analytic theory of such spaces; in particular, our result implies that GCBA spaces satisfy the infinity Poincaré inequality and an essential assumption in the theory of Lipschitz-Volume rigidity.

math.DG↗

arXiv2025

A characterization of snowflakes via rectifiability

We prove a generalization of Tyson-Wu's characterization of metric spaces biLipschitz equivalent to snowflakes to every metric space, by removing compactness, doubling and embeddability assumptions. We also characterize metric spaces that are biLipschitz equivalent to a snowflake in terms of the absence of non-trivial metric $1$-currents in every ultralimit, or equivalently in terms of purely $1$-unrectifiability of every ultralimit. Finally, we discuss some applications and examples.

math.MG↗

arXiv2024

Otal-Peigné's Theorem for Gromov-hyperbolic spaces

We extend the classical Otal-Peigné's Theorem to the class of proper, Gromov-hyperbolic spaces that are line-convex. Namely, we prove that when a group acts discretely and virtually freely by isometries on a metric space in this class then its critical exponent equals the topological entropy of the geodesic flow of the quotient metric space. We also show examples of proper, Gromov-hyperbolic spaces that are not line-convex and for which the statement fails.

math.MG↗

arXiv2024

Sobolev spaces via chains in metric measure spaces

We define the chain Sobolev space on a possibly non-complete metric measure space in terms of chain upper gradients. In this context, $\varepsilon$-chains are a finite collection of points with distance at most $\varepsilon$ between consecutive points. They play the role of discrete versions of curves. Chain upper gradients are defined accordingly and the chain Sobolev space is defined by letting the size parameter $\varepsilon$ going to zero. In the complete setting, we prove that the chain Sobolev space is equal to the classical notions of Sobolev spaces in terms of relaxation of upper gradients or of the local Lipschitz constant of Lipschitz functions. The proof of this fact is inspired by a recent technique developed by Eriksson-Bique. In the possible non-complete setting, we prove that the chain Sobolev space is equal to the one defined via relaxation of the local Lipschitz constant of Lipschitz functions, while in general they are different from the one defined via upper gradients along curves. We apply the theory developed in the paper to prove equivalent formulations of the Poincaré inequality in terms of pointwise estimates involving $\varepsilon$-upper gradients, lower bounds on modulus of chains connecting points and size of separating sets measured with the Minkowski content in the non-complete setting. Along the way, we discuss the notion of weak $\varepsilon$-upper gradients and asymmetric notions of integral along chains.

math.MG↗

arXiv2024

Bishop-Jones' Theorem and the ergodic limit set

For a proper, Gromov-hyperbolic metric space and a discrete, non-elementary, group of isometries, we define a natural subset of the limit set at infinity of the group called the ergodic limit set. The name is motivated by the fact that every ergodic measure which is invariant for the geodesic flow on the quotient metric space is concentrated on geodesics with endpoints belonging to the ergodic limit set. We refine the classical Bishop-Jones' Theorem proving that the packing dimension of the ergodic limit set coincides with the critical exponent of the group.

math.DS↗

arXiv2024

Finiteness of CAT(0) group actions

We prove some finiteness results for discrete isometry groups $Γ$ of uniformly packed CAT$(0)$-spaces $X$ with uniformly bounded codiameter (up to group isomorphism), and for CAT$(0)$-orbispaces $M = Γ\backslash X$ (up to equivariant homotopy equivalence or equivariant diffeomorphism); these results generalize, in nonpositive curvature, classical finiteness theorems of Riemannian geometry. As a corollary, the order of every torsion subgroup of $Γ$ is bounded above by a universal constant only depending on the packing constants and the codiameter. The main tool is a splitting theorem for sufficiently collapsed actions: namely we show that if a geodesically complete, packed, CAT$(0)$-space admits a discrete, cocompact group of isometries with sufficiently small systole then it necessarily splits a non-trivial Euclidean factor.

math.GR↗

arXiv2024

Convergence and collapsing of CAT$(0)$-lattices

We study the theory of convergence for CAT$(0)$-lattices (that is groups $Γ$ acting geometrically on proper, geodesically complete CAT$(0)$-spaces) and their quotients (CAT$(0)$-orbispaces). We describe some splitting and collapsing phenomena, explaining precisely how these action can degenerate to a possibly non-discrete limit action. Finally, we prove a compactness theorem for the class of compact CAT$(0)$-homology orbifolds, and some applications: an isolation result for flat orbispaces and an entropy-pinching theorem.

math.MG↗

arXiv2024

A geometric approach to Poincaré inequality and Minkowski content of separating sets

The goal of this paper is to continue the study of the relation between the Poincaré inequality and the lower bounds of Minkowski content of separating sets, initiated in our previous work [Caputo, Cavallucci: Poincaré inequality and energy of separating sets, arXiv 2401.02762]. A new shorter proof is provided. An intermediate tool is the study of the lower bound of another geometric quantity, called separating ratio. The main novelty is the description of the relation between the infima of the separating ratio and the Minkowski content of separating sets. We prove a quantitative comparison between the two infima in the local quasigeodesic case and equality in the local geodesic one. No Poincaré assumption is needed to prove it. The main tool employed in the proof is a new function, called the position function, which allows in a certain sense to fibrate a set in boundaries of separating sets. We also extend the proof to measure graphs, where due to the combinatorial nature of the problem, the approach is more intuitive. In the appendix, we revise some classical characterizations of the p-Poincaré inequality, by proving along the way equivalence with a notion of p-pencil that extends naturally the definition for p = 1.

math.MG↗

arXiv2024

Discrete groups of packed, non-positively curved, Gromov hyperbolic metric spaces

We prove a quantitative version of the classical Tits' alternative for discrete groups acting on packed Gromov-hyperbolic spaces supporting a convex geodesic bicombing. Some geometric consequences, as uniform estimates on systole, diastole, algebraic entropy and critical exponent of the groups, will be presented. Finally we will study the behaviour of these group actions under limits, providing new examples of compact classes of metric spaces.

math.MG↗

arXiv2024

Poincaré inequality and energy of separating sets

We study geometric characterizations of the Poincaré inequality in doubling metric measure spaces in terms of properties of separating sets. Given a couple of points and a set separating them, such properties are formulated in terms of several possible notions of energy of the boundary, involving for instance the perimeter, codimension type Hausdorff measures, capacity, Minkowski content and approximate modulus of suitable families of curves. We prove the equivalence within each of these conditions and the $1$-Poincaré inequality.

math.MG↗

arXiv2023

GH-convergence of CAT$(0)$-spaces: stability of the Euclidean factor

We prove that if a sequence of geodesically complete CAT$(0)$-spaces $X_j$ with uniformly cocompact discrete groups of isometries converges in the Gromov-Hausdorff sense to $X_\infty$, then the dimension of the maximal Euclidean factor splitted off by $X_\infty$ and $X_j$ is the same, for $j$ big enough. In other words, no additional Euclidean factors can appear in the limit.

math.MG↗

arXiv2023

The Metric Completion of the Space of Vector-Valued One-Forms

The space of full-ranked one-forms on a smooth, orientable, compact manifold (possibly with boundary) is metrically incomplete with respect to the induced geodesic distance of the generalized Ebin metric. We show a distance equality between the induced geodesic distances of the generalized Ebin metric on the space of full-ranked one-forms and the corresponding Riemannian metric defined on each fiber. Using this result we immediately have a concrete description of the metric completion of the space of full-ranked one-forms. Additionally, we study the relationship between the space of full-ranked one-forms and the space of all Riemannian metrics, leading to quotient structures for the space of Riemannian metrics and its completion.

math.DG↗

arXiv2023

A GH-compactification of CAT$(0)$-groups via totally disconnected, unimodular actions

We give a detailed description of the possible limits in the equivariant-Gromov-Hausdorff sense of sequences $(X_j,G_j)$, where the $X_j$'s are proper, geodesically complete, uniformly packed, CAT$(0)$-spaces and the $G_j$'s are closed, totally disconnected, unimodular, uniformly cocompact groups of isometries. We show that the class of metric quotients $G/X$, where $X$ and $G$ are as above, is compact under Gromov-Hausdorff convergence. In particular it is a geometric compactification of the class of locally geodesically complete, locally compact, locally CAT$(0)$-spaces with uniformly packed universal cover and uniformly bounded diameter.

math.MG↗

arXiv2023

Ahlfors regular conformal dimension and Gromov-Hausdorff convergence

We prove that the Ahlfors regular conformal dimension is upper semicontinuous with respect to Gromov-Hausdorff convergence when restricted to the class of uniformly perfect, uniformly quasi-selfsimilar metric spaces. Moreover we show the continuity of the Ahlfors regular conformal dimension in case of limit sets of discrete, quasiconvex-cocompact group of isometries of uniformly bounded codiameter of $δ$-hyperbolic metric spaces under equivariant pointed Gromov-Hausdorff convergence of the spaces.

math.MG↗

arXiv2022

Thin actions on CAT(0) spaces

We study groups of isometries of packed, geodesically complete, CAT$(0)$-spaces for which the systole at every point is smaller than a universal constant depending only on the packing, deducing strong rigidity results. We show that if a space as above has some negative curvature behaviour then it cannot support a thin action: this generalizes the classical Margulis Lemma to a broader class of spaces.

math.MG↗