[Paper Review] Veech groups, irrational billiards and stable abelian differentials

Ferrán Valdez|ArXiv.org|May 11, 2009

Mathematical Dynamics and Fractals2 references5 citations

TL;DR

This paper investigates Veech groups of non-compact flat surfaces arising from irrational polygonal billiards and irreducible stable abelian differentials. It proves that for irrational billiards, the Veech group lies in SO(2,R) and is infinitely generated when angles are free of resonances; for stable abelian differentials with non-parallel residues, the group is finite, while with parallel residues, it lies within a specific subgroup N and may be discrete or equal to N.

ABSTRACT

We describe Veech groups of flat surfaces arising from irrational angled polygonal billiards or irreducible stable abelian differentials. For irrational polygonal billiards, we prove that these groups are non-discrete subgroups of SO(2,R) and we calculate their rank.

Motivation & Objective

• To characterize Veech groups of flat surfaces derived from irrational polygonal billiards, which are non-compact and of infinite genus.
• To extend the notion of Veech groups to stable abelian differentials on irreducible nodal Riemann surfaces at the boundary of the Deligne-Mumford compactification.
• To construct uncountably many flat surfaces homeomorphic to the Loch Ness monster with infinitely generated Veech groups in SO(2,R).
• To analyze the structure of Veech groups based on the geometric and dynamical properties of singularities and holonomy vectors.
• To determine conditions under which Veech groups are finite, discrete, or equal to a maximal subgroup N in SO(2,R).

Proposed method

• Introduces the concept of a tame flat surface with finite or infinite angle singularities, ensuring well-behaved metric completion.
• Applies the Katok-Zemljakov construction to unbounded polygons with irrational angles to produce flat surfaces homeomorphic to the Loch Ness monster.
• Defines Veech groups as the group of derivatives of orientation-preserving affine homeomorphisms on non-compact surfaces.
• Uses holonomy vectors and rotation matrices of the form $\begin{pmatrix} \cos(2\lambda_j\pi) & -\sin(2\lambda_j\pi) \\ \sin(2\lambda_j\pi) & \cos(2\lambda_j\pi) \end{pmatrix}$ to generate the Veech group.
• Applies group-theoretic criteria: if the set of residues of a stable differential is not parallel in $\mathbf{C} \simeq \mathbf{R}^2$, the Veech group is finite.
• Establishes that when all residues are parallel, the Veech group lies within $N = \langle \{ \begin{smallmatrix} 1 & s \\ 0 & t \end{smallmatrix} \mid t>0, s\in\mathbb{R} \}, -\mathrm{Id} \rangle$, and is either discrete or equal to $N$.

Experimental results

Research questions

• RQ1 What is the structure of the Veech group for a flat surface constructed from an irrational polygonal billiard?
• RQ2 Under what conditions is the Veech group of a stable abelian differential on a nodal Riemann surface finite or infinite?
• RQ3 Can one construct uncountably many flat surfaces with infinitely generated Veech groups in SO(2,R)?
• RQ4 How does the resonance-free condition on angles affect the rank and structure of the Veech group?
• RQ5 What determines whether the Veech group of a stable differential is discrete or equal to the maximal subgroup N?

Key findings

• For any irrational polygonal billiard with at least one irrational angle, the Veech group is a non-discrete subgroup of SO(2,R).
• The Veech group is infinitely generated when the sequence of interior angles $\lambda_j\pi$ is free of resonances.
• If the residues of a stable abelian differential are not parallel in $\mathbf{C}$, the Veech group is finite.
• When all residues are parallel, the Veech group lies within the subgroup $N = \langle \{ \begin{smallmatrix} 1 & s \\ 0 & t \end{smallmatrix} \mid t>0, s\in\mathbb{R} \}, -\mathrm{Id} \rangle$.
• The Veech group equals $N$ if all holonomy vectors are horizontal; otherwise, it is a discrete subgroup of $N$.
• An uncountable family $\mathcal{S}_n = \{S_i\}_{i\in I}$ of flat surfaces exists, each homeomorphic to the Loch Ness monster, with infinitely generated Veech groups in SO(2,R).