arXiv research

No primitive Teichmüller curves found in Prym(2,2).

It has been known since 1981 that if one fixes an orientable surface $S$ of genus $g$, then there is a real number $λ_{min,g} > 1$ that is the dilatation of a pA diffeomorphism of $S$, and every other pA diffeomorphism of $S$ has dilatation $\geq λ_{min,g}$. We will show how a little-known theorem about digraphs gives …

It has been known since 1981 that if one fixes an orientable surface $S$ of genus $g$, then there is a real number $λ_{min,g} > 1$ that is the dilatation of a pA diffeomorphism of $S$, and every other pA diffeomorphism of $S$ has dilatation $\geq λ_{min,g}$. We will show how a little-known theorem about digraphs gives …

Given a Prym-Teichmüller curve in $\mathcal{M}_3$, this note provides an invariant that sorts the cusp prototypes of Lanneau and Nguyen by component. This can be seen as an analogue of McMullen's genus $2$ spin invariant, although the source of this invariant is different. Moreover, we describe the Galois action on the…

In this paper we study the minimum dilatation pseudo-Anosov mapping classes coming from fibrations over the circle of a single 3-manifold, the mapping torus for the "simplest pseudo-Anosov braid". The dilatations that arise include the minimum dilatations for orientable mapping classes for genus g=2,3,4,5,8 as well as …

Prym-Teichmüller curves $W_D(4)$ constitute the main examples of known primitive Teichmüller curves in the moduli space $\mathcal{M}_3$. We determine, for each non-square discriminant $D>1$, the number and type of orbifold points in $W_D(4)$. These results, together with the formulas of Lanneau-Nguyen and Möller for th…

Estimates KVol on surfaces with geometric constraints.

The paper calculates the size of origami orbit graphs in complex surfaces.

Study intersections of curves on translation surfaces, focusing on regular polygons and their Teichmüller disks.

Moduli spaces of Abelian and quadratic differentials are stratified by multiplicities of zeroes; connected components of the strata correspond to ergodic components of the Teichmuller geodesic flow. It is known that the strata are not necessarily connected; the connected components were recently classified by M. Kontse…

For each discriminant $D>1$, McMullen constructed the Prym-Teichmüller curves $W_D(4)$ and $W_D(6)$ in $\mathcal{M}_{3}$ and $\mathcal{M}_{4}$, which constitute one of the few known infinite families of geometrically primitive Teichmüller curves. In the present paper, we determine for each $D$ the number and type of or…

Classifies components of k-differentials and their orbit closures.

Classifies components of strata of k-differentials on Riemann surfaces.

New findings on translation lengths in Teichmüller and curve graphs for pseudo-Anosovs.