[Paper Review] Complete periodicity of Prym eigenforms

Erwan Lanneau, Duc-Manh Nguyen|arXiv (Cornell University)|Jan 4, 2013

Mathematical Dynamics and Fractals15 references9 citations

TL;DR

This paper establishes complete periodicity for Prym eigenforms in genus 3, 4, and 5, proving that all homological directions are algebraically periodic and all regular closed geodesics lie in completely periodic directions. Using a novel approach to periodic foliations via linear involutions, the authors show that the limit set of the Veech group is either empty, a point, or the full circle at infinity, leading to new examples of translation surfaces with infinitely generated Veech groups of the first kind.

ABSTRACT

This paper deals with Prym eigenforms which are introduced previously by McMullen. We prove several results on the directional flow on those surfaces, related to complete periodicity (introduced by Calta). More precisely we show that any homological direction is algebraically periodic, and any direction of a regular closed geodesic is a completely periodic direction. As a consequence we draw that the limit set of the Veech group of every Prym eigenform in some Prym loci of genus 3,4, and 5 is either empty, one point, or the full circle at infinity. We also construct new examples of translation surfaces satisfying the topological Veech dichotomy. As a corollary we obtain new translation surfaces whose Veech group is infinitely generated and of the first kind.

Motivation & Objective

• To establish complete periodicity for Prym eigenforms in genus 3, 4, and 5, extending results from genus 2.
• To characterize the limit set of the Veech group for Prym eigenforms in specific strata, showing it is either empty, a single point, or the full circle at infinity.
• To construct new examples of translation surfaces satisfying the topological Veech dichotomy that are not rationally covered by Veech surfaces.
• To demonstrate the existence of infinitely many Prym eigenforms with infinitely generated Veech groups of the first kind.

Proposed method

• Introduce a new approach to periodic foliations using linear involutions to analyze directional flows on translation surfaces.
• Use the Sah-Arnoux-Fathi (SAF) invariant to define algebraic periodicity and relate it to complete periodicity.
• Analyze cylinder decompositions in homological directions and exploit symmetry under the Prym involution τ to locate saddle connections and geodesic segments.
• Construct explicit examples of translation surfaces in Prym(1,1,2), Prym(4,4)even, and Prym(1,1,4) with specific period coordinates and real multiplication by a quadratic order.
• Apply Dehn twists to perturb directions and show that unstable periodic directions are dense near any regular closed geodesic direction.
• Use the fact that a Fuchsian group is of the first kind if its limit set is the full circle at infinity, and show that non-lattice Veech groups must be infinitely generated.

Experimental results

Research questions

• RQ1 Does every homological direction on a Prym eigenform exhibit algebraic periodicity, i.e., does the SAF invariant vanish?
• RQ2 Is every direction containing a regular closed geodesic completely periodic for Prym eigenforms in genus 3, 4, and 5?
• RQ3 What are the possible limit sets of the Veech group for Prym eigenforms in the strata Q(8), Q(−1,2,3), and Q(−13,1,2)?
• RQ4 Can new examples of non-Veech translation surfaces satisfy the topological Veech dichotomy?
• RQ5 Do there exist infinitely many Prym eigenforms with infinitely generated Veech groups of the first kind?

Key findings

• All homological directions on a Prym eigenform are algebraically periodic, meaning the SAF invariant vanishes.
• All directions containing a regular closed geodesic are completely periodic, implying the flow is either periodic or uniquely ergodic.
• For any Prym eigenform in the strata Q(8), Q(−1,2,3), or Q(−13,1,2), the limit set of the Veech group is either empty, a single point, or the full circle at infinity.
• There exist infinitely many translation surfaces in Prym(1,1,2), Prym(4,4)even, and Prym(1,1,4) whose Veech groups are infinitely generated and of the first kind.
• Explicit constructions of surfaces in Prym(1,1,2) with incommensurable cylinder moduli demonstrate non-Veech behavior and infinite generation of the Veech group.
• The topological Veech dichotomy holds for all quadratic differentials in Q(8) and Q(−1,2,3) stabilized by a pseudo-Anosov map, providing first examples not arising as ramified covers of Veech surfaces.