Research

The space of co-geodesic currents of a hyperbolic group (joint work with Eduardo Reyes), in preparation 

We introduce "co-geodesic currents," for a hyperbolic group G, a family of G-invariant measures on hyperplanes at infinity for a Gromov hyperbolic group G, generalizing geodesic currents from surface groups. Co-geodesic currents naturally arise from geometric actions of G on CAT(0) cube complexes, actions on real trees with bounded-bactracking property or quasi-fuchsian representations.  We establish a natural intersection pairing between co-geodesic and geodesic currents, extending Bonahon’s intersection number for surface groups in the case of classical geodesic currents. Moreover, we show that every non-trivial co-geodesic current defines a pseudo-metric space on the space of triplets of G with a measured wall structure.

We deduce that the existence of non-trivial co-geodesic currents imply lack of property (T).

Bounded backtracking property and geodesic currents (joint work with Misha Kapovich), in preparation 

Abstract: We study actions of hyperbolic groups on R-trees satisfying the bounded-bactracking property (BBT), saying that geodesics are uniformly close to quasi-geodesics. We show that under BBT,  the stable length extends to geodesic currents. Then we prove that, for one-ended groups, every minimal small action satisfies BBT. In particular, small actions of one ended hyperbolic groups have stable length extending continuously.

Injectivity and continuity of the intersection form in infinite type (joint work with Tommaso Cremaschi), (draft here)

Abstract: We study geodesic currents on infinite-type surfaces with respect to the weak^* and uniform weak^* topologies. We investigate two fundamental properties of the intersection pairing which are classical in the finite-type setting: injectivity of the marked length spectrum and continuity of the intersection form. We show that geodesic currents on infinite-type surfaces are still determined by their intersection numbers with closed curves, extending Otal's marked length spectrum rigidity argument to this setting. In contrast, we prove that the intersection pairing is not continuous in general for either the weak^* or uniform weak^* topology by constructing explicit counterexamples. We then introduce a uniform decay condition at infinity under which continuity is restored. As an application, we adapt Kerckhoff's convexity argument to obtain a Nielsen realization result in the infinite-type setting.

Geodesic currents of coarse negative curvature (joint work with Meenakshy Jyothis), ArXiv: https://arxiv.org/abs/2605.14469  (draft available) 

Abstract:   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 thermodynamic 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. 

The intersection dual of geodesic currents (joint work with Dylan P. Thurston) ArXiv: https://arxiv.org/abs/2605.04031 (draft available)

Abstract: 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 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.

A geometric correspondence for reparameterizations of geodesic flows (joint work with Stephen Cantrell and Eduardo Reyes). ArXiv: https://arxiv.org/abs/2605.02585

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.

We further establish continuity of the Bowen--Margulis--Sullivan geodesic current map on the moduli space of metrics, as well as a Bowen-type description of these currents as limits of sums of appropriately normalized atomic geodesic currents. For surface groups, we apply this continuity result to show that the Bowen--Margulis--Sullivan map restricts to a topological embedding on Hitchin components (up to contragradient involution) when equipped with their Hilbert lengths.

Horoboundary and rigidity of filling geodesic currents (joint work with Meenakshy Jyothis), 2026 (Submitted) ArXiv: 2601.02059 . 

Abstract: 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.

Comparing hyperbolic and extremal length for short curves (joint work with Franco Vargas Pallete), 2019 (Submitted) ArXiv: https://arxiv.org/abs/1911.09078. 

Abstract: 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.

Growth tightness and genericity for word metrics from injective spaces (joint work with Lihuang Ding and Abdul Zalloum), (Accepted in Compositio Mathematica) ArXiv: https://arxiv.org/abs/2407.02378

Abstract: 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.  

Dual spaces of geodesic currents (joint work with Luca de Rosa), (Published in Journal of Topology, 18 (2025), no. 4, e70045) ArXiv: https://arxiv.org/abs/2211.05164 

Abstract: 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 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.  

The extremal length systole of the Bolza surface (joint work with Maxime Fortier Bourque and Franco Vargas Pallete), (Published in Annales Henri Lebesgue, Vol 7, 2024, 1409-1455) ArXiv: https://arxiv.org/abs/2105.03871

Abstract: We prove that the extremal length systole of genus two surfaces attains a strict local maximum at the Bolza surface, where it takes the value square root of 2.

Here are some slides for a talk I gave in the Pacific Dynamics seminar, in May 2021. 

Volume bounds for the canonical lift complement of a random geodesic (joint work with Tommaso Cremaschi, Yannick Krifka, Franco Vargas Pallete), (Published in Trans. Amer. Math. Soc. Ser. B 10 (2023), 988-1038) ArXiv: https://arxiv.org/abs/2108.08419

Abstract: Given a filling primitive geodesic loop in a closed hyperbolic surface one obtains a hyperbolic three-manifold as the complement of the loop'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 loops. 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. 

From curves to currents (joint work with Dylan P. Thurston), (Published in Forum of Mathematics, Sigma (2021), Vol. 9:e77 1–52 ) Open Access Link   ArXiv Link (with Correction (Forum of Mathematics, Sigma (2025), Vol. 13:e16 1–4 ).

Abstract: We extend a broad family of functionals defined on weighted multi-curves to the space of geodesic currents. This subsumes many previous extension results of functionals of curves to currents, but also provides with extensions of functionals that hadn't been considered before, such as extremal length.

Here's a poster based on this pre-print I presented at the LMS Interactions between Geometry, Dynamics and Group Theory, at Bristol University, in January 2020.

Here are some slides on a talk on this I gave at the Wisconsin-Milwaukee Topology Seminar, November 2020, emphasizing the application to extremal length counting, or a video of a talk I gave at the MSRI program on Random and Arithmetic structures of Fall 2021, emphasizing the connections to higher rank.