Veech groups of covers of the Chamanara surface

TL;DR

This work studies finite-area infinite translation surfaces arising as covers of the Chamanara surface $X$, focusing on when their Veech groups have finite index in $\Gamma(X)$. By describing normal abelian covers via monodromy vectors $h$ in $G^{\mathbb{Z}}$ and exploiting the 'everything descends' property, the authors reduce the finite-index problem to fixed-point dynamics of a hyperbolic element $H$ acting on $h$-vectors, yielding a concrete criterion: $\Gamma(Y_h)$ has finite index in $\Gamma(X)$ iff $h$ is a fixed point of $H^n$ for some $n$. The degree-2 case is analyzed through weakly $n$-periodic vectors and Schreier graphs of Striezel and Kranz types, showing that every free group occurs as a projective Veech group of some finite-area infinite translation surface. The paper also identifies devious covers with infinite index and studies the topology of covers, proving that for any degree $d$ there are covers with $d$ ends, with a complete description in the degree-$2$ case. Collectively, the results connect monodromy, group actions, and Teichmüller dynamics to classify large Veech groups and realize free groups in this geometric context.

Abstract

We study finite abelian covers of the Chamanara surface, an example of a finite-area infinite translation surface with interesting dynamics and a large Veech group. Specifically, the Veech group of the Chamanara surface is a virtually free group on two generators. We characterize when finite abelian covers have large Veech groups themselves, namely when their Veech group has finite index in that of the Chamanara surface. For degree-2 covers, we provide a detailed analysis of these finite-index Veech groups. As an application, we prove that every free group arises as the projective Veech group of a finite-area infinite translation surface.

Figures (7)

• Figure 1: The Chamanara surface, also known as the baker's map surface. We identify, using translations, segments that are parallel and have the same length.
• Figure 2: Chamanara surface with a cylinder decomposition of slope $1$ (left) and of slope $2$ (right).
• Figure 3: The first generators of $\pi_1(X)$.
• Figure 4: The action of $H^{-1}$ on the Chamanara surface.
• Figure 5: The images of $\beta_1$, $\alpha_1$, and $\alpha_2$ under $H^{-1}$.

Theorems & Definitions (70)

• Theorem 1: Realizing $F_n$ as Veech group
• Theorem 2: Characterization of finite index ($d=2$)
• Theorem 3: Characterization of finite index
• Theorem 4: Topology of covers
• Remark 1: Translation structure on covers
• Lemma 1: Translations of the universal cover are deck transformations
• proof
• Proposition 1: Everything descends property
• proof
• Corollary 1: All Veech group elements of the cover are lifts