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Dídac Martínez-Granado

Publications and source records attributed to Dídac Martínez-Granado.

11 recordsLinked to original sources
arXiv2026

Large-scale rank and geometry of the Lipschitz metric on geodesic currents

For a closed orientable surface of genus at least 2, we study the large-scale geometry of the (symmetrized) Lipschitz metric on its space of projective filling geodesic currents. We show that the rank and, hence, asymptotic dimension of the space of filling geodesic currents is infinite. This gives the first example of a natural proper space associated to a closed surface, with a metrically proper isometric action of the mapping class group, that has infinite asymptotic dimension. In contrast, we show that any finite simplex of geodesic currents has rank uniformly bounded above by $8g-9$. We then show a sufficient criterion for its rank and asymptotic dimension to be equal to $8g-9$ and construct examples satisfying it, thus showing the bound is sharp. We further prove that Teichmüller space equipped with the Lipschitz metric has rank equal to $3g-3$. In particular, the space of projective filling geodesic currents and the Teichmüller space are not quasi-isometric.

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arXiv2026

Comparing hyperbolic and extremal lengths for shortest curves

We give a lower bound for the widths of the collars of certain short partial pants decomposition of the surface. Then we apply this to obtain upper bounds of the renormalized volume of certain Schottky manifolds in terms of the hyperbolic length of compressible curves.

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arXiv2026

A geometric correspondence for reparameterizations of geodesic flows

For any non-elementary, torsion-free hyperbolic group, we provide a correspondence between the left-invariant Gromov-hyperbolic metrics on the group that are quasi-isometric to a word metric, and continuous reparameterizations of the associated Mineyev's flow space. From this correspondence, we produce the first examples of continuous reparameterizations of geodesic flows on negatively curved manifolds with all periodic orbits having integer lengths. For surface and free groups, this also yields isometric actions on Gromov-hyperbolic spaces on which loxodromic elements are precisely the non-simple elements. Key ingredients in our proof are an analysis of the geometry of Mineyev's flow space (such as the metric-Anosov property recently proven by Dilsavor), and the density of Green metrics in the moduli space of (symmetric) metrics on the group.

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arXiv2026

Geodesic currents of coarse negative curvature

Strong hyperbolicity is a coarse notion of negative curvature, stronger than Gromov hyperbolicity, that includes all CAT(-k) metrics for k positive and allows the use of dynamical techniques available in negative curvature, such as thermodynamical formalism. We prove that the subset of geodesic currents whose dual pseudometric is strongly hyperbolic is dense in the space of geodesic currents. The proof combines an elementary finite-cover argument with a characterization of strong hyperbolicity in terms of boundary data for pseudometrics dual to geodesic currents. In contrast, we show that currents arising from non-positively curved metrics on the surface are not dense. As a consequence, we construct infinitely many pairwise non-roughly-isometric invariant strongly hyperbolic geodesic metrics on the universal cover of the surface which are not CAT(0). Finally, we establish correlation counting results for the associated length spectra.

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arXiv2026

The intersection dual of geodesic currents

Geodesic currents on closed hyperbolic surfaces are measures on the unit tangent bundle invariant under geodesic flow and orientation reversal. Every geodesic current induces a dual function on curves via the geometric intersection pairing. It is natural to ask which curve functions are dual to geodesic currents, that is, which arise as intersection functionals of a geodesic current. In this paper we give a purely axiomatic and combinatorial characterization of curve functionals dual to geodesic currents. This yields a new definition of geodesic currents as curve functionals or, equivalently, as functions on surface groups, without reference to measures or flows. More precisely, we show that a function on curves arises as the geometric intersection pairing with a geodesic current if and only if it is additive under disjoint union and satisfies a simple \emph{smoothing} property: it is non-increasing under surgery of essential crossings. As applications, we obtain new axiomatic characterizations of measured laminations and hyperbolic length functions, and new descriptions of small surface group actions on real trees, including a concise proof of a classical theorem of Skora. We also provide a unified framework for dual geodesic currents arising from metric structures and generalized cross-ratios, including those associated with certain Anosov representations. Our approach subsumes all previously known constructions of dual geodesic currents and yields broad new families of examples.

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arXiv2026

Horoboundary and rigidity of filling geodesic currents

We endow the space of projective filling geodesic currents on a closed hyperbolic surface with a natural asymmetric metric extending Thurston's asymmetric metric on Teichmüller space, as well as analogous metrics arising from Hitchin representations. More generally, we show that this metric extends beyond surface groups and geodesic currents, and encompasses metrics associated with Anosov representations of Gromov hyperbolic groups. We identify the horofunction compactification of the space of projective filling currents equipped with this metric with the space of projective geodesic currents. As a consequence, we obtain a rigidity result: the metric spaces of projective filling geodesic currents associated with closed surfaces of distinct genera are not isometric.

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arXiv2025

Growth tightness and genericity for word metrics from injective spaces

Mapping class groups are known to admit geometric (proper, cobounded) actions on injective spaces. Starting with such an action, and relying only on geometric arguments, we show that all finite generating sets resulting from taking large enough balls in the respective injective space yield word metrics where pseudo-Anosov maps are exponentially generic. We also show that growth tightness holds true for the Cayley graphs corresponding to these finite generating sets, providing a positive answer to a question by Arzhantseva, Cashen and Tao.

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arXiv2025

Dual spaces of geodesic currents

Every geodesic current on a hyperbolic surface has an associated dual space. If the current is a lamination, this dual embeds isometrically into a real tree. We show that, in general, the dual space is a Gromov hyperbolic metric tree-graded space, and express its Gromov hyperbolicity constant in terms of the geodesic current. In the case of geodesic currents with no atoms and full support, such as those coming from certain higher rank representations, we show the duals are homeomorphic to the surface. We also analyze the completeness of the dual and the properties of the action of the fundamental group of the surface on the dual. Furthermore, we compare two natural topologies in the space of duals.

math.GT↗
arXiv2024

From curves to currents

Many natural real-valued functions of closed curves are known to extend continuously to the larger space of geodesic currents. For instance, the extension of length with respect to a fixed hyperbolic metric was a motivating example for the development of geodesic currents. We give a simple criterion on a curve function that guarantees a continuous extension to geodesic currents. The main condition of our criterion is the smoothing property, which has played a role in the study of systoles of translation lengths for Anosov representations. It is easy to see that our criterion is satisfied for almost all the known examples of continuous functions on geodesic currents, such as non-positively curved lengths or stable lengths for surface groups, while also applying to new examples like extremal length. We use this extension to obtain a new curve counting result for extremal length.

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arXiv2022

Volume bounds for the canonical lift complement of a random geodesic

Given a filling primitive geodesic curve in a closed hyperbolic surface one obtains a hyperbolic three-manifold as the complement of the curve's canonical lift to the projective tangent bundle. In this paper we give the first known lower bound for the volume of these manifolds in terms of the length of generic curves. We show that estimating the volume from below can be reduced to a counting problem in the unit tangent bundle and solve it by applying an exponential multiple mixing result for the geodesic flow.

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