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On the Ends of groups and the Veech groups of infinite-genus surfaces

Camilo Ramírez Maluendas

TL;DR

This work shows that any finitely generated subgroup $G$ of $GL_{+}(2,\\mathbb{R})$ without contracting elements can be realized as the Veech group of a tame translation surface $S$ of infinite genus, constructed via a refined PSV procedure. The authors connect the ends structure of $S$ to the ends of $G$: if $G$ is finite, $S$ has as many ends as $G$, each with infinite genus; if $G$ is infinite, ${\\rm Ends}(S)$ splits into a closed part homeomorphic to ${\\rm Ends}(G)$ and a countable, dense, open subset, providing a precise topological correspondence between the ends of the group and the ends of the surface. The construction uses decorated and buffer surfaces, a Cayley-graph-based puzzle, and gluing along parallel markings to ensure tameness and to realize the desired Veech group. These results advance the problem of realizing symmetry groups of infinite-genus translation surfaces and establish a concrete link between gluing procedures, end spaces, and linear groups acting on the plane, with potential implications for dynamics on such surfaces.

Abstract

In this paper, we study the PSV construction, which provides a step by step method for obtaining tame translation surfaces with a suitable Veech group. In addition, we modify slightly this construction, and for each finitely generated subgroup $G<{\rm GL}_{+}(2,\mathbb{R})$ without contracting elements, we produce a tame translation surface $S$ with infinite genus such that its Veech group is $G$. Furthermore, the ends space of $S$ can be written as $\mathcal{B}\sqcup \mathcal{U}$, where $\mathcal{B}$ is homeomorphic to the ends space of the group $G$, and $\mathcal{U}$ is a countable, discrete, dense, and open subset of the ends space of $S$.

On the Ends of groups and the Veech groups of infinite-genus surfaces

TL;DR

This work shows that any finitely generated subgroup of without contracting elements can be realized as the Veech group of a tame translation surface of infinite genus, constructed via a refined PSV procedure. The authors connect the ends structure of to the ends of : if is finite, has as many ends as , each with infinite genus; if is infinite, splits into a closed part homeomorphic to and a countable, dense, open subset, providing a precise topological correspondence between the ends of the group and the ends of the surface. The construction uses decorated and buffer surfaces, a Cayley-graph-based puzzle, and gluing along parallel markings to ensure tameness and to realize the desired Veech group. These results advance the problem of realizing symmetry groups of infinite-genus translation surfaces and establish a concrete link between gluing procedures, end spaces, and linear groups acting on the plane, with potential implications for dynamics on such surfaces.

Abstract

In this paper, we study the PSV construction, which provides a step by step method for obtaining tame translation surfaces with a suitable Veech group. In addition, we modify slightly this construction, and for each finitely generated subgroup without contracting elements, we produce a tame translation surface with infinite genus such that its Veech group is . Furthermore, the ends space of can be written as , where is homeomorphic to the ends space of the group , and is a countable, discrete, dense, and open subset of the ends space of .

Paper Structure

This paper contains 11 sections, 11 theorems, 43 equations, 4 figures.

Table of Contents

  1. Ends
  2. Ends of a surface
  3. Ends of a group
  4. Tame translation surfaces
  5. Saddle connection and markings
  6. Veech group
  7. Proof of Theorem \ref{['theorem.0.2']}
  8. PSV construction
  9. We employ PSV construction to obtain the surface $S$
  10. The surface $S$ is a tame translation surface and its Veech group is the subgroup $G<{\rm GL}_{+}(2,\mathbb{R})$
  11. Ends space of the surface $S$

Key Result

Theorem 1

Given a finitely generated subgroup $G$ of ${\rm GL}_{+}(2,\mathbb{R})$ without contracting elements. Then there exists a tame translation surface $S$ whose Veech group is $G$. The ends space ${\rm Ends}(S)$ of $S$ satisfies:

Figures (4)

  • Figure 1: Gluing markings.
  • Figure 2: Buffer surface $S(Id,h_j)$.
  • Figure 3: Decorated surface $S_{{\rm dec}}$.
  • Figure 4: Image of $\beta_j$.

Theorems & Definitions (26)

  • Theorem 1
  • Corollary 1
  • Definition 1: Fre
  • Theorem 2: Ray
  • Remark 1
  • Theorem 3: Classification of non-compact surfaces KerIan
  • Definition 2: PSul
  • Remark 2: SPE
  • Proposition 1: Loh
  • Theorem 4: Fre1Hopf
  • ...and 16 more