research

Given two hyperbolic surfaces and a homotopy class of maps between them, Thurston proved that there always exists a representative minimizing the Lipschitz constant. While not unique, these minimizers are rigid along a geodesic lamination. In this paper, we investigate what happens in the complement of that lamination. To do this, we introduce deflations, certain optimal maps to trees which can be used to obstruct optimal maps between surfaces. Using a smooth version of the orthogeodesic foliation of the first author and Farre, we also construct many new families of optimal maps, showing that the obstructions coming from deflations are essentially the only ones.

We compute the asymptotic number of cylinders, weighted by their area to any non-negative power, on any cyclic branched cover of any generic translation surface in any stratum. Our formulas depend only on topological invariants of the cover and number-theoretic properties of the degree: in particular, the ratio of the related Siegel–Veech constants for the locus of covers and for the base stratum component is independent of the number of branch values. One surprising corollary is that this ratio for area3 Siegel–Veech constants is always equal to the reciprocal of the the degree of the cover. A key ingredient is a classification of the connected components of certain loci of cyclic branched covers.

We show that for any surface of genus at least 3 equipped with any choice of framing, the graph of non-separating curves with winding number 0 with respect to the framing is hierarchically hyperbolic but not Gromov hyperbolic. We also describe how to build analogues of the curve graph for marked strata of abelian differentials that capture the combinatorics of their boundaries, analogous to how the curve graph captures the combinatorics of the augmented Teichmüller space. These curve graph analogues are also shown to be hierarchically, but not Gromov, hyperbolic.

We address a conjecture of Mirzakhani about the statistical behavior of certain expanding families of “twist tori” in the moduli space of hyperbolic surfaces, showing that they equidistribute to a certain Lebesgue-class measure along almost all sequences. We also identify a number of other expanding families of twist tori whose limiting distributions are mutually singular to Lebesgue.

In a previous paper, the authors extended Mirzakhani’s (almost-everywhere defined) measurable conjugacy between the earthquake and horocycle flows to a measurable bijection. In this one, we analyze the continuity properties of this map and its inverse, proving that both are continuous at many points and in many directions. This lets us transfer measure convergence between the two systems, allowing us to pull back results from Teichmu ̈ller dynamics to deduce analogous statements for the earthquake flow.

Any pair of intersecting cylinders on a translation surface is “coherent,” in that the geometric and algebraic intersection numbers of their core curves are equal (up to sign). In this paper, we investigate when a pair of multicurves can be simultaneously realized as the core curves of cylinders on some translation surface. Our main tools are surface topology and the “flat grafting” deformation introduced by Ser-Wei Fu.

By work of Jenkins and Strebel, given a Riemann surface X and a simple closed multi-curve a on it, there exists a unique quadratic differential q on X whose horizontal foliation is equivalent to a. We study the distribution of the critical graphs of these differentials in the moduli space of metric ribbon graphs as the extremal length of the multi-curves goes to infinity, showing they equidistribute to the Kontsevich measure regardless of the initial choice of X.

Cutting a hyperbolic surface X along a simple closed multi-geodesic results in a hyperbolic structure on the complementary subsurface. We study the distribution of the shapes of these subsurfaces in moduli space as boundary lengths go to infinity, showing that they equidistribute to the Kontsevich measure on a corresponding moduli space of metric ribbon graphs. In particular, random subsurfaces look like random ribbon graphs, a law which does not depend on the initial choice of X. This result strengthens Mirzakhani’s famous simple closed multi-geodesic counting theorems for hyperbolic surfaces.

We extend Mirzakhani’s conjugacy between the earthquake and horocycle flows to a bijection, demonstrating conjugacies between these flows on all strata and exhibiting an abundance of new ergodic measures for the earthquake flow. The structure of our map indicates a natural extension of the earthquake flow to an action of the the upper-triangular subgroup P < SL₂R and we classify the ergodic measures for this action as pullbacks of affine measures on the bundle of quadratic differentials. Our main tool is a generalization of the shear coordinates of Bonahon and Thurston to arbitrary measured laminations.

This paper investigates the relationship between strata of abelian differentials and various mapping class groups afforded by means of the topological monodromy representation. Building off of prior work of the authors, we show that the fundamental group of a stratum surjects onto the subgroup of the mapping class group which preserves a fixed framing of the underlying Riemann surface, thereby giving a complete characterization of the monodromy group. In the course of our proof we also show that these “framed mapping class groups” are finitely generated (even though they are of infinite index) and give explicit generating sets.

For g ≥ 5, we give a complete classification of the connected components of strata of abelian differentials over Teichmüller space, establishing an analogue of a theorem of Kontsevich and Zorich in the setting of marked translation surfaces. Building on work of the first author, we find that the non-hyperelliptic components are classified by an invariant known as an r–spin structure. This is accomplished by computing a certain monodromy group valued in the mapping class group. To do this, we determine explicit finite generating sets for all r–spin stabilizer subgroups of the mapping class group, completing a project begun by the second author. Some corollaries in flat geometry and toric geometry are obtained from these results.

Let Σ be a surface with either boundary or marked points, equipped with an arbitrary framing. In this note we determine the action of the associated “framed mapping class group” on the homology of Σ relative to its boundary (respectively marked points), describing the image as the kernel of a certain crossed homomorphism related to classical spin structures. Applying recent work of the authors, we use this to describe the monodromy action of the orbifold fundamental group of a stratum of abelian differentials on the relative periods.

This paper describes connected components of the strata of holomorphic abelian differentials on marked Riemann surfaces with prescribed degrees of zeros. Unlike the case for unmarked Riemann surfaces, we find there can be many connected components, distinguished by roots of the cotangent bundle of the surface. In the course of our investigation we also characterize the images of the fundamental groups of strata inside of the mapping class group. The main techniques of proof are mod r winding numbers and a mapping class group–theoretic analogue of the Euclidean algorithm.

The bounce spectrum of a polygonal billiard table is the collection of all bi-infinite sequences of edge labels corresponding to billiard trajectories on the table. We give methods for reconstructing from the bounce spectrum of a polygonal billiard table both the cyclic ordering of its edge labels and the sizes of its angles. We also show that it is impossible to reconstruct the exact shape of a polygonal billiard table from any finite collection of finite words from its bounce spectrum.

We present explicit geometric decompositions of the hyperbolic complements of alternating k- uniform tiling links, which are alternating links whose projection graphs are k-uniform tilings of S2, E2, or H2. A consequence of this decomposition is that the volumes of spherical alternating k-uniform tiling links are precisely twice the maximal volumes of the ideal Archimedean solids of the same combinatorial description, and the hyperbolic structures for the hyperbolic alternating tiling links come from the equilateral realization of the k-uniform tiling on H2. In the case of hyperbolic tiling links, we are led to consider links embedded in thickened surfaces Sg × I with genus g ≥ 2 and totally geodesic boundaries. We generalize the bipyramid construction of Adams to truncated bipyramids and use them to prove that the set of possible volume densities for all hyperbolic links in Sg × I, ranging over all g ≥ 2, is a dense subset of the interval [0, 2voct], where voct ≈ 3.66386 is the volume of the ideal regular octahedron.

The volume density of a hyperbolic link is defined as the ratio of hyperbolic volume to crossing number. We study its properties and a closely-related invariant called the determinant density. It is known that the sets of volume densities and determinant densities of links are dense in the interval [0,v_{oct}]. We construct sequences of alternating knots whose volume and determinant densities both converge to any x in [0,v_{oct}]. We also investigate the distributions of volume and determinant densities for hyperbolic rational links, and establish upper bounds and density results for these invariants.