Research

My research interests are in the fields low-dimensional geometry and topology, hyperbolic geometry, and Bers-Teichmüller theory. I use analytical tools, such as geodesic currents or extremal length, to study the geometric structure of spaces stemming from curves on hyperbolic surfaces or 3-manifolds.

The papers listed as "in preparation" are stable versions of preprints that still require some edits to be published. They can be shared on request.

In preparation

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

Abstract: We study a family of actions of hyperbolic groups on R-trees and show that their stable length extends continuously to geodesic currents. The result applies in particular to small actions of one ended hyperbolic groups.

Geodesic currents and intersections (joint work with Dylan P. Thurston)

Abstract: We give a characterization of functions defined on the space of closed curves that come from intersection numbers with a geodesic current. The key condition is that the function must be non-increasing under smoothing essential crossings.

The configuration of lifts of loops defining a box of geodesics. We define a pre-measure on these boxes.This family of boxes is a semi-ring in the sense of measure theory.

Pre-prints

Dual spaces of geodesic currents (joint work with Luca de Rosa), 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 dual space of a geodesic current is a hyperbolic tree-graded space

The extremal length systole of the Bolza surface (joint work with Maxime Fortier Bourque and Franco Vargas Pallete), (submitted) 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. 

The quadratic differential that realizes the extremal length of the Bolza surface

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

Abstract: We give a uniform lower bound for the widths of the collars of a short pair of pants decomposition. We derive from it a uniform linear upper bound of the extremal length of curves in a short pair of pants decomposition in terms of the hyperbolic length. 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 and a lower bound in terms of the genus for the total extremal length of a pair of pants decomposition of a random hyperbolic surface.

Any short geodesic has an orthogonal geodesic of universal width.

Publications

Volume bounds for the canonical lift complement of a random geodesic (joint work with Tommaso Cremaschi, Yannick Krifka, Franco Vargas Pallete), (Accepted at Transactions of AMS) 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. 

The random closed geodesic constructed by taking long trajectories of the geodesic flow and closing them off by a short segment.

From curves to currents (joint work with Dylan P. Thurston), (published in Forum of Mathematics, Sigma) Open Access Link   ArXiv Link

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.

The family of functionals on curves considered satisfies, among other conditions, that it decreases under smoothings.