research

I am interested in the interplay between geometry, topology and group theory, which puts me somewhere in the nexus of geometric group theory, geometric topology, and low-dimensional topology. I am particularly interested in the relationship between mapping class groups, Teichmüller geometry, and flat surfaces.

papers and preprints:

Shear-shape cocycles for measured laminations and ergodic theory of the earthquake flow
(with James Farre)
Submitted (2021). (pdf)(arXiv)

abstract

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.


Framed mapping class groups and the monodromy of strata of Abelian differentials
(with Nick Salter)
Submitted (2020). (pdf)(arXiv)

abstract

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.

Higher spin mapping class groups and strata of Abelian differentials over Teichmüller space
(with Nick Salter)
Adv. Math. 389: Paper no. 107926 (2021). (pdf) (arXiv)(journal)

abstract

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.

Relative homological representations of framed mapping class groups
(with Nick Salter)
B. Lond. Math. Soc. 53 (1): 204-219 (2021). (pdf)(arXiv)(journal)

abstract

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.

Connected components of strata of Abelian differentials over Teichmüller space
Comment. Math. Helv. 95 (2): 361–420 (2020). (pdf) (arXiv)(journal)

abstract

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.


How to hear the shape of a billiard table.
(with Solly Coles, Diana Davis, Justin Lanier, and Andre Oliveira)
Submitted (2018). (arXiv)

abstract

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.


Generalized bipyramids and hyperbolic volume of alternating tiling links.
(with Colin Adams and Nat Mayer)
Topology Appl. 271:1–28, 2020. (pdf) (arXiv) (journal)

abstract

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.

Volume and determinant densities of hyperbolic rational links.
(with Colin Adams, Xinyi Jiang, Alex Kastner, Greg Kehne, Nat Mayer, and Mia Smith)
J. Knot Theory Ramifications 26 (1), 2017. (arXiv) (journal)

abstract

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.


Conjugacy geodesics in Coxeter groups
(with S. Hermiller and T. Susse)
Senior Thesis.