Papers $\def\Z{\mathbb Z} \def\Q{\mathbb Q} \def\Torelli{\mathcal I}$

We construct moduli spaces of complex affine and dilation surfaces. Using ideas of Veech, we show that the the moduli space of affine surfaces with fixed genus and with cone points of fixed complex order is a holomorphic affine bundle over the moduli space of Riemann surfaces. Similarly, the moduli space of dilation surfaces is a covering space of the moduli space of Riemann surfaces. We classify the connected components of the moduli space of dilation surfaces and show that any component is an orbifold K(G,1) where G is the framed mapping class group of Calderon-Salter.

The orbit closure of the unfolding of every rational right and isosceles triangle is computed and the asymptotic number of periodic billiard trajectories in these triangles is deduced. This follows by classifying all orbit closures of rank at least two in hyperelliptic components of strata of Abelian and quadratic differentials. Additionally, given a fixed set of angles, the orbit closure of the unfolding of all unit area rational parallelograms, isosceles trapezoids, and right trapezoids outside of a discrete set is determined.

We classify the GL(2,R)-invariant subvarieties M in strata of Abelian differentials for which any two M-parallel cylinders have homologous core curves. This answers a question of Mirzakhani and Wright. As a corollary we show that outside of an explicit list of exceptions, if M is a GL(2,R)-invariant subvariety, then the Kontsevich-Zorich cocycle has nonzero Lyapunov exponents in the symplectic orthogonal of the projection of the tangent bundle of M to absolute cohomology.

We classify GL(2,R) orbit closures of translation surfaces of rank at least half the genus plus 1.

We classify point markings over genus two Abelian differentials and show that exotic examples of orbit closures discovered by Kumar-Mukamel and Eskin-McMullen-Mukamel-Wright are unique. Applications to determining holomoprhic sections over Hilbert modular curves are given.

We classify a natural collection of GL(2,R)-invariant subvarieties which includes loci of double covers as well as the orbits of the Eierlegende-Wollmilchsau, Ornithorynque, and Matheus-Yoccoz surfaces. This is motivated in part by a forthcoming application to another classification result, the classification of "high rank" invariant subvarieties. We also give new examples, which negatively resolve two questions of Mirzakhani and Wright, clarify the complex geometry of Teichmuller space, and illustrate new behavior relevant to the finite blocking problem.

We classify the periodic points on the regular n-gon and double n-gon translation surfaces and deduce consequences for the finite blocking problem on rational triangles that unfold to these surfaces.

We introduce and study diamonds of GL(2,R)-invariant subvarieties of Abelian and quadratic differentials, which allow us to recover information on an invariant subvariety by simultaneously considering two degenerations, and which provide a new tool for the classification of invariant subvarieties. We investigate the surprisingly rich range of situations where the two degenerations are contained in "trivial" invariant subvarieties. Our main results will be applied in forthcoming work to classify large collections of invariant subvarieties; the statement of those results will not involve diamonds, but their proofs will use them as a crucial tool.

The set of directions from a quadratic differential that diverge on average under Teichmuller geodesic flow has Hausdorff dimension exactly equal to one-half.

We prove strong finiteness results on the orbits of marked translation surfaces under geodesic flow and apply the work to the finite blocking problem in rational billiards.

The Hausdorff dimension, starting at any translation surface, of the set of directions in which Birkhoff or Oseledets averages deviate a definite amount from the expected limit is strictly less than one. The dimension of the set of directions that diverge on average is bounded above by 1/2 and this bound is sharp. This strengthens results of Masur and Chaika-Eskin.

We classify orbit closures of marked translation surfaces when the unmarked translation surface is generic with respect to the GL(2,R)-action. The illumination and finite blocking problems on generic translation surfaces are resolved as corollaries.

Every GL(2,R)-orbit in hyperelliptic components of strata of abelian differentials in genus greater than two is either closed, dense, or contained in a locus of branched covers.

We show that all affine invariant submanifolds of complex dimension greater than three in hyperelliptic components of strata of abelian differentials are branched covering constructions, i.e. every translation surface in the affine invariant submanifold covers a translation surface in a lower genus hyperelliptic component of a stratum of abelian differentials. This result implies finiteness of algebraically primitive Teichmuller curves in all hyperelliptic components for genus greater than two. A classification of all GL(2,R) orbit closures in hyperelliptic components of strata (up to computing connected components and up to finitely many nonarithmetic rank one orbit closures) is provided conditional on the sparsity conjecture.

A finite group is called a B-group if all its irredundant generating sets are the same size. We classify B-groups with trivial Frattini subgroup and deduce a structure theorem for B-groups as a corollary.