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

TL;DR

This paper develops a method to construct infinite-genus translation surfaces with prescribed Veech groups, linking the geometric properties of the surface to the algebraic properties of the group.

Contribution

It introduces a modified PSV construction that produces tame translation surfaces with specified finitely generated Veech groups and describes their ends space structure.

Findings

Constructed surfaces have Veech groups matching given subgroups without contracting elements.

The ends space of the surfaces decomposes into parts related to the group and a dense open subset.

The method generalizes previous constructions to infinite-genus 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 $S$.

Equations

$$\mathrm{Ends}(S)={\mathrm{Ends}}_{\infty }(S)=\mathcal{B}\bigsqcup \mathcal{U},$$

$$U^{\ast }:=\{[U_n{]}_{n\in \mathbb{N}}\in \mathrm{Ends}(X)|U_j\subset U\text{for some }j\in \mathbb{N}\}.$$

$${\varphi }_{\alpha }\circ {\varphi }_{\beta }^{-1}:{\varphi }_{\beta }(U_{\alpha }\cap U_{\beta })\subset {\mathbb{R}}^2\to {\varphi }_{\alpha }(U_{\alpha }\cap U_{\beta })\subset {\mathbb{R}}^2,$$

$$\rho :{\mathrm{Aff}}_{+}(S)\to {\mathrm{GL}}_{+}(2,\mathbb{R}),$$

$$S:=\bigcup\limits_{g\in G}S_{g}\bigg{/}\sim,$$

$${\check{M}}^j=\{{\check{m}}_i^j=(4i{e}_1,(4i+1)e_1):\forall i\in \mathbb{N}\}\text{ and }L=\{l_i=((4i+2)e_1,(4i+3)e_1):\forall i\in \mathbb{N}\}.$$

$$L^{{}^{\prime }}=\{l_i^{{}^{\prime }}=((2i+1)e_2,e_1+(2i+1)e_2):\forall i\in \mathbb{N}\},$$

$$h_j{\check{m}}^{-j}=(2e_2,e_1+2e_2).$$

$$S(Id,h_j),$$

$$M^j$$

$$M$$

$$m^{-1}=(x_1{e}_1+y_1{e}_2,x_1{e}_1+h_1^{-1}{e}_1+y_1{e}_2)$$

$$m^{-n}=(x_n{e}_1+y_n{e}_2,x_n{e}_1+h_n^{-1}{e}_1+y_n{e}_2),$$

$$S_{\mathrm{dec}},$$

$$S_g^{{}^{\prime }}:=S_g\setminus \{g{h}_j{\check{m}}^{-j},g{m}^{-j}:j\le J\}\subset S_g,$$

$$\mathrm{Ends}(S)={\mathrm{Ends}}_{\infty }(S)=\mathcal{B}\bigsqcup \mathcal{U},$$

$$S\setminus \bigcup _{g\in G}\partial S_g^{{}^{\prime }}=\bigcup _{g\in G}{S}_g^{{}^{\prime }}$$

$$S\setminus C=\bigcup _{g\in G}(S_g\setminus C_g),$$

$$\mathcal{U}:=\left\{[U(g{)}_n{]}_{n\in \mathbb{N}}\in \mathrm{Ends}(S):g\in G\right\}\subset \mathrm{Ends}(S).$$

$$\tilde{W}:=\mathrm{Int}\left(\bigcup _{g\in G\cap (W\cup \partial W)}{\overline{S^{{}^{\prime }}}}_g\right)\subset S.$$

$$[W_n{]}_{n\in \mathbb{N}}\mapsto [{\tilde{W}}_n{]}_{n\in \mathbb{N}}.$$

$$g\mapsto {\overline{\text{0}}}_g.$$

$$f_g\circ {\beta }_j:[0,1]\to S,$$

$$i:\mathrm{Cay}(G,H)\hookrightarrow S,$$

$$Z_n=\mathrm{Int}\left(\bigcup _{g\in G(n)}{\overline{S^{{}^{\prime }}}}_g\right)\subset S.$$

Full text

Partially supported by UNIVERSIDAD NACIONAL DE COLOMBIA, SEDE MANIZALES, Project Hermes 47019

Introduction

Geometrically, an end of a space is a point at infinity. In the manuscript [Fre], H. Freudenthal introduced the concept of ends, and explored some of its applications in group theory. Hence, one can define ${\rm Ends}(G)$ the end space of a finitely generated group $G$ as the ends space of the Cayley graph ${\rm Cay}(G,H)$, where $H$ is a generating set of $G$ [Fre1, Hopf]. Subsequently, the mathematician B. Kerékjártó studied the ends of the orientable surfaces and introduced the classification of non-compact orientable surfaces; which determines the topological type of any orientable surface $S$ by its genus $g(S)\in\mathbb{N}\cup\{\infty\}$, and two closed subsets ${\rm Ends}_{\infty}(S)\subset{\rm Ends}(S)$ of the Cantor set. These subsets are called the ends space of $S$ and the ends of $S$, having (infinite) genus (see [Ker, Ian]).

The translation surfaces appear naturally in the study of several problems in Dynamical systems (see [HuSch, KMS]), Teichmüller theory (see [KoZo, Moll]), Riemann surfaces (see [MasTab, Zor]), among others. We focus on the so-called tame translation surfaces. Using the charts of translate surface $S$, one can pull back the standard Riemannian metric on $\mathbb{R}^{2}$ to equip the surface $S$ with a flat Riemannian metric $\mu$. This flat metric induces a distance map $d$ on $S$. The translation surface $S$ is tame if for each point $x$ in the metric completion $\hat{S}$ with respect to $d$ there is a neighborhood $U_{x}\subset\hat{S}$ isometric to either: an open of the Euclidean plane or, an open of a ramification point of a (finite or infinite) cyclic branched covering of the unit disk. Each compact translation surface is tame. Nevertheless, there are examples of translation surfaces that are not tame [Chamanara]. From the historical point of view, these objects appeared by first time in the manuscript [FoKe] of R. H. Fox and R. B. Kershner. There, the authors described the unfolding process for rational billiard, which associates to each billiard $\phi_{P}$ coming from an Euclidean compact polygon $P\subset\mathbb{E}^{2}$, a tame translate surface $S_{P}$, and a projection map $\pi_{p}:S_{p}\to\phi_{P}$, sending each geodesic of $S_{P}$ onto a billiard trajectory of $\phi_{P}$. Several authors have studied such surface, see e.g., [ZeKa, VaWe, FrCo].

During the 1989s, W. A. Veech [Vee], associated to each translation surface an affine subgroup $\Gamma$ of ${\rm GL}(2,\mathbb{R})$, which was subsequently called the Veech group, and he proved that: if the Veech group $\Gamma$ of a compact translation surface $S$ is a lattice, it means, $\Gamma$ is a Fuchsian group such that the quotient space $\mathbb{H}^{2}/\Gamma$ has finite hyperbolic area, then the behavior of the geodesic flow on $S$ has dynamical properties similar to those described in the theorem of Weyl for the geodesic flow on the torus [KMS, Subsection 1.4]. This result is well-known as the Veech’s Dichotomy. In [Voro] one can find a new proof of this theorem. The study of the Veech groups has attracted the attention of several researchers e.g., [PasLan, Hoop, Fin].

The Veech group of a compact translation surface is a Fuchsian group [Voro]. One might ask: which groups are Veech groups of a non-compact tame translate surface? In the document [PSV], P. Przytycki, G. Weitze-Schmithüsen, and F. Valdez described all the subgroups of ${\rm GL}_{+}(2,\mathbb{R})$, which may be Veech group of a tame translation surface. The result states the following:

Theorem 1** ([PSV]*Theorem 1.1).**

Let $G<{\rm GL}_{+}(2,\mathbb{R})$ be the Veech group of a tame translation surface $S$. Then one of the following holds:

$G$* is countable and without contracting elements, it means, $G$ is disjoint from the set $\{A\in{\rm GL}_{+}(2,\mathbb{R}):\|Av\|<\|v\|\text{ for all }v\in\mathbb{R}^{2}\setminus\{\textbf{0}\}\}$, where $\|\,\|$ is the euclidean norm on $\mathbb{R}^{2}$, or* 2. 2.

$G$* is conjugated to $P:=\left\{\begin{pmatrix}1&t\\ 0&s\end{pmatrix}\hskip 2.84526pt:\hskip 2.84526ptt\in\mathbb{R}\text{ and }\hskip 2.84526pts\in\mathbb{R}^{+}\right\},\hskip 2.84526pt\text{or}$* 3. 3.

$G$* is conjugated to $P^{\prime}<{\rm GL_{+}(2,\mathbb{R})}$, the subgroup generated by $P$ and $\rm-Id$, or* 4. 4.

$G$* is equal to ${\rm GL_{+}(2,\mathbb{R})}$.*

The only tame translate surfaces with Veech group ${\rm GL}_{+}(2,\mathbb{R})$ are the cyclic branched coverings of the Euclidean plane [PSV, Lemma 3.2]. The realization problem for Fuchsian group as Veech group of a compact translate surface [HuTho, p.524] naturally leads to the following question: Which subgroup of ${\rm GL}_{+}(2,\mathbb{R})$ satisfying either: 1, 2 or 3 of the above theorem are realized as Veech groups of a non-compact tame translation surface? For the case, $G$ is a subgroup of ${\rm GL}_{+}(2,\mathbb{R})$ without contracting elements, in the above mentioned work [PSV], the authors have introduced an interesting process, which we call the PSV construction, whose purpose it is to obtain a tame translate surface homeomorphic to the Loch Ness monster with Veech group $G$. In [RamVal], the author jointly with F. Valdez varied lightly the PSV construction and obtained a lot of surfaces of infinite genus with tame translation structure and Veech group $G$, where $G$ is a subgroup of ${\rm GL}(2,\mathbb{R})$ without contracting elements. We have also explored and slightly modified the PSV construction. The main contribution of this manuscript is to built a nice tame translate surface $S$ with Veech group $G$ such that its topological type is related to the ends space of $G$. We can now state the main result.

Theorem 2**.**

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 $G$ and, its topological type satisfies:

If $G$ is finite, then the surface $S$ has ${\rm card}(G)$ ends and each one of them have infinite genus. 2. 2.

If $G$ is not finite, then the ends space of $S$ can be represented by

[TABLE]

where $\mathcal{B}$ is a closed subset of ${\rm Ends}(S)$ homeomorphic to ${\rm Ends}(G)$, and $\mathcal{U}$ is a countable discrete dense open subset of ${\rm Ends}(S)$.

As the end space of a finitely generated group has either: zero, one, two or infinitely many ends [Fre1, Hopf], then it is immediate the following result.

Corollary 1**.**

The topological type of the tame translation surface $S$ is either:

The ends space ${\rm Ends}(S)$ is homeomorphic to the ordinal number $\omega+1$, if the ends space ${\rm Ends}(G)$ has one end. 2. 2.

The ends space ${\rm Ends}(S)$ is homeomorphic to the ordinal number $\omega\cdot 2+1$, if the ends space ${\rm Ends}(G)$ has two ends. 3. 3.

The ends space ${\rm Ends}(S)$ has a subset homeomorphic to the Cantor set, if the ends space ${\rm Ends}(G)$ has infinitely many ends.

In Section 1, we collect the principal tools to understand the classification of non-compact surfaces theorem. Also, we explore the concept of ends on groups. In Section 2, we provide an introduction to the theory of tame translate surface. Moreover, we state the Veech group and the PSV construction. Finally, the section 3 is dedicated to the proof of our main result.

1. Ends

In this section we shall introduce the space of ends of a topological space $X$ in the most general context. Also, we shall explore the classification of non-compact and orientable surfaces theorem from the ends space of a surface. Finally, we shall discuss the concept of ends on groups.

1.1. The ends space of a topological space ([Fre, 1. Kapitel.])

Let $X$ be a locally compact, locally connected, connected, and Hausdorff space, and let $(U_{n})_{n\in\mathbb{N}}$ be an infinite nested sequence $U_{1}\supset U_{2}\supset\ldots$ of non-empty connected open subsets of $X$, such that the following hold:

For each $n\in\mathbb{N}$ the boundary $\partial U_{n}$ of $U_{n}$ is compact, 2. 2.

The intersection $\cap_{n\in\mathbb{N}}\overline{U_{n}}=\emptyset$, and 3. 3.

For each compact $K\subset X$ there is $m\in\mathbb{N}$ such that $K\cap U_{m}=\emptyset$.

Two nested sequences $(U_{n})_{n\in\mathbb{N}}$ and $(U^{\prime}_{n})_{n\in\mathbb{N}}$ are equivalent if for each $n\in\mathbb{N}$ there exist $j,k\in\mathbb{N}$ such that $U_{n}\supset U^{\prime}_{j}$, and $U^{\prime}_{n}\supset U_{k}$. The corresponding equivalence classes of these sequences are called the ends of $X$, and the set of all ends of $X$ is denote by ${\rm Ends}(X)$. The ends space of $X$ is the topological space having the ends of $X$ as elements, and endowed with the following topology: for every non-empty open subset $U$ of $X$ such that its boundary $\partial U$ is compact, we define

[TABLE]

Then we take the set of all such $U^{\ast}$, with $U$ open with compact boundary of $X$, as a basis for the topology of ${\rm Ends}(X)$.

Theorem 3** ([Ray]*Theorem 1.5).**

The space ${\rm Ends}(X)$ is Hausdorff, totally disconnected, and compact.

1.2. Ends of a surface

A surface S is a connected $2$-manifold without boundary (which may or may not be closed). Throught this manscript, we shall only consider orientable surfaces. The surface $S$ is said to be planar if all of its compact subsurface are of genus zero, and the genus of the surface $S$ is the maximum of the genera of its compact subsurfaces.

An end $[U_{n}]_{n\in\mathbb{N}}$ of a surface $S$ is called planar if there is $l\in\mathbb{N}$ such that the open subset $U_{l}\subset S$ is planar. Hence, we define the subset ${\rm Ends}_{\infty}(S)$ of ${\rm Ends}(S)$ conformed by all ends of $S$, which are not planar (ends having infinity genus). It follows directly from the definition that ${\rm Ends}_{\infty}(S)$ is a closed subset of ${\rm Ends}(S)$ (see [Ian, p. 261]).

Theorem 4** (Classification of non-compact surfaces, [Ker]*§7. Erster Abschnitt [Ian]*Theorem 1).**

Two surfaces $S_{1}$ and $S_{2}$ having the same genus are topological equivalent if and only if there exists a homeomorphism $f:{\rm Ends}(S_{1})\to{\rm Ends}(S_{2})$ such that $f({\rm Ends}_{\infty}(S_{1}))={\rm Ends}_{\infty}(S_{2})$.

1.3. The Loch Ness monster ([Val]*Definition 2)

The Loch Ness Monster is the unique, up to homeomorphism, surface of infinite genus and exactly one end. From the historical point of view as shown in [AyC], this nomenclature is due to A. Phillips and D. Sullivan [PSul].

Remark 1** ([SPE]*§5.1., p. 320).**

The surface $S$ has $n$ end if and only if for all compact subset $K\subset S$ there is a compact $K^{{}^{\prime}}\subset S$ such that $K\subset K^{{}^{\prime}}$ and $S\setminus K^{{}^{\prime}}$ is conformed by $n$ connected components.

1.4. Ends of a group

Given a generating set $H$ (closed under invertion) of a group $G$, the Cayley graph of $G$ with respect to the generating set $H$ is the graph ${\rm Cay}(G,H)$ whose vertices are the elements of $G$ and, there is an edge with ends points $g_{1}$ and $g_{2}$ if and only if there is an element $h\in H$ such that $g_{1}h=g_{2}$. Along this paper, the Cayley graph ${\rm Cay}(G,H)$ will be the geometric realization of an abstract graph (see [Diestel2]*p.226).

When the set $H$ is finite, the Cayley graph ${\rm Cay}(G,H)$ is locally compact, locally connected, connected, Hausdorff space. In this case, we define the ends space of $G$ as ${\rm Ends}(G):={\rm Ends}({\rm Cay}(G,H))$.

Proposition 1** ([Loh]*Proposition 8.2.5).**

Let $G$ be a finitely generated group. The ends space of the Cayley graph of $G$ does not depend on the choice of the finite genereting set.

Theorem 5** ([Fre1]3. Kapitel., [Hopf]§3. Die Enden abstrakter Gruppen).**

Let $G$ be a finitely generated group. Then $G$ has either zero, one, two or infinitely many ends.

2. Tame translation surfaces

An atlas $\mathcal{A}=\{(U_{\alpha},\phi_{\alpha})\}_{\alpha\in I}$ on the surface $S$ is called translation if for any choice of charts $(U_{\alpha},\phi_{\alpha})$ and $(U_{\beta},\phi_{\beta})$ belong to $\mathcal{A}$ sucht that $U_{\alpha}\cap U_{\beta}\neq\emptyset$, then the associated transition map

[TABLE]

is locally the restriction of a translation. A translation structure on $S$ is a maximal translation atlas on the surface. If the surface $S$ admits a translation structure, then it will be called a translation surface. From the charts of the translation atlas on the surface $S$ we can lift the Euclidean metric of $\mathbb{R}^{2}$ to flat Riemannian metric $\mu$ to $S$. We denote by $\hat{S}$ the metric completion of $S$ with respect to the Riemannian distance. As a consequence of the Uniformization Theorem [Abi]*p: 580, the only complete translation surfaces $S=\hat{S}$ are the Euclidean plane, the torus and the cylinder (see [FarKra, p. 193]).

2.1. Tame translation surface ([PSV]*Definition 2.2)

A translation surface $S$ is called tame if for every point $x\in\widehat{S}$ there exists a neighborhood $U_{x}\subset\hat{S}$ isometric to either:

Some open of the Euclidean plane,

• 2.

An open of the ramification point of a (finite or infinite) cyclic branched covering of the unit disk in the Euclidean plane.

In the later case, the point $x$ is called finite cone angle singularity of angle $2n\pi$, if the cyclic covering has finite order $n\in\mathbb{N}$. Contrary, if the cyclic covering is infinite, then the point is called infinite cone angle singularity,

We denote by $\Sigma(S)_{\rm fin}$ the set conformed by all finite cone angle singularities of the translation surface $S$ and $\Sigma(S)_{\rm inf}$ the set conformed by all infinite cone angle singularities of $S$. A point in the set $\Sigma(S)=\Sigma(S)_{\rm fin}\cup\Sigma(S)_{\rm inf}$ is called cone angle singularity of $S$ or just cone point. Let us remark that $\hat{S}\setminus\Sigma(S)_{\rm inf}$ is a surface. Throughout this article we will use the term the topological type of the tame translation surface $S$ to refer to the topological type of the surface $\hat{S}\setminus\Sigma(S)_{\rm inf}$.

A saddle connection of a tame translation surface $S$ is a geodesic interval joining two cone points and not having cone points in its interior. A saddle connection $\gamma$ is locally a straight line in the direction $[\theta]$, for any $[\theta]\in\mathbb{R}/2\pi\mathbb{Z}$. Then we can associate to each saddle connection $\gamma$ two holonomy vectors $\{v,-v\}\subset\mathbb{R}^{2}$ such that they are in the direction $[\theta]$ and $[-\theta]$, respectively. Moreover, the norm of the holonomy vectors $v$ and $-v$ are equal to the length of $\gamma$ with respect to the flat Riemannian metric $\mu$ on $S$.

A mark on the tame translate surface $S$ is a finite length geodesic not having cone points inside it. Similarly, as the saddle connected case we can associate to each mark two holonomy vectors $\{v,-v\}\subset\mathbb{R}^{2}$. Then, two marks are parallel if their respective holonomy vectors are also parallel. It does not matter if the marks are on different surfaces (see [PSV, Definition 3.4]).

2.2. Gluing marks ([RamVal]*Definition 1.2)

Given $m_{1}$ and $m_{2}$ two parallel marks having the same length on translation surfaces $S_{1}$ and $S_{2}$, respectively, then we cut $S_{1}$ and $S_{2}$ along $m_{1}$ and $m_{2}$ respectively, thus we turn $S_{1}$ and $S_{2}$ into the surface with boundary $\tilde{S}_{1}$ and $\tilde{S}_{2}$, respectively, each one of their boundaries are formed by two straight segments, respectively. Now, we consider the union $\tilde{S}_{1}\cup\tilde{S}_{2}$ and identify (glue) such (four) segments using translations to obtain a connected tame translation surface $S$ (see Figure 1). The relation of these segments will be denoted as $m_{1}\sim_{\text{glue}}m_{2}$ and will be called: the operation of gluing the marks $m_{1}$ and $m_{2}$. Then the surface $S$ will be written in the form $S=(S_{1}\cup S_{2})/m_{1}\sim_{\text{glue}}m_{2}$, and we say that $S$ is obtained from $S_{1}$ and $S_{2}$ by regluing along $m_{1}$ and $m_{2}$.

An affine diffeomorphism of a tame translation surface $S$ is a homeomorphism $f:\widehat{S}\to\widehat{S}$, which maps cone points to cone points, and is an affine map in the local coordinates of the translation atlas on $S$. We denote by ${\rm Aff}_{+}(S)$ the group conformed by all affine orientation preserving diffeomorphism of the tame translation surface $S$. If we choose an element $f\in{\rm Aff}_{+}(S)$, using the translation structure on $S$, we hold that $df(p)$ the differential of $f$ at any point $p\in S$ is a constant matrix $A$ belongs to ${\rm GL}_{+}(2,\mathbb{R})$. Hence, we have a well defined the group homomorphism

[TABLE]

where $\rho(f)$ is the differential matrix of $f$.

2.3. The Veech group ([Vee]*p. 556)

The image of $\rho$, that we denote by $\Gamma(S)$, is called the Veech group of $S$.

The group ${\rm GL}_{+}(2,\mathbb{R})$ acts on the set conformed by all translation surfaces by post composition on charts. More precisely, this action sends the couple $(g,S)$ to the translation surface $S_{g}$, which is called the affine copy of $S$, and whose translation structure is obtained by postcomposing every chart on $S$ by the affine transformation associated to the matrix $g$. Further, this action defines an affine diffeomorphism $f_{g}:S\to S_{g}$, such that $df_{g}(p)$ the differential of $f_{g}$ at any point $p\in S$ is the matrix $g$.

2.4. PSV construction

For each countable subgroup $G$ of ${\rm GL}_{+}(2,\mathbb{R})$ without contracting elements, in [PSV]*4. Countable Veech group, P. Przytycki, G. Weitze-Schmithüsen and F. Valdez have described a way to build a tame translation surface homeomorphic to the Loch Ness monster with Veech group $G$. We call this method the PSV construction. From the topological point of view, this process is as follows:

Step 1.

We obtain a suitable tame Loch Ness monster $S_{{\rm dec}}$, from copies of the Euclidean plane and a cyclic branched covering of the Euclidean plane, which are appropriately attached using the gluing marks. The surface $S_{\rm dec}$ is called decorated. Let $H$ be a (finite or infinite) generating set of the group $G$ and let $\omega$ be the ordinal number of the well-ordered set $\{1,2,\ldots\}$ in its natural order. Then we can choose an ordinal number $J\leq\omega$ so that $H=\{h_{j}:j\leq J\}$ and, for each element $j\in J$ we draw on $S_{{\rm dec}}$ two infinite families of (suitable) marks $h_{j}\check{M}^{-j}=\{h_{j}\check{m}^{-j}_{i}:\forall i\in\mathbb{N}\}$ and $M^{-j}=\{m^{-j}_{i}:\forall i\in\mathbb{N}\}$111In [RamVal] the authors built decorated surfaces with topological type different to the Loch Ness monster..

Step 2.

For each element $g\in G$, we take the affine copy $S_{g}$ of the decorated surface $S_{{\rm dec}}$. Then we denote by $gh_{j}\check{M}^{-j}=\{gh_{j}\check{m}^{-j}_{i}:\forall i\in\mathbb{N}\}$ and $gM^{-j}=\{gm^{-j}_{i}:\forall i\in\mathbb{N}\}$ (respectively) the two families of marks on $S_{g}$, which are given by the image of the families $h_{j}\check{M}^{-j}$ and $M^{-j}$ (respectively) using the diffeomorphism $f_{g}:S_{\rm dec}\to S_{g}$. Thus, we define the puzzle associated to the triplet $(1,G,H)$ given by $\mathfrak{P}(1,G,H):=\{S_{g}:g\in G\}$ (see [RamVal, Definition 3.1]). The symbol $1$ means that the decorated surface has only one end.

Step 3.

We define the assembled surface to the puzzle $\mathfrak{P}(1,G,H)$ ([RamVal, Definition 3.1]) as follows

[TABLE]

such that $\sim$ is the equivalent relation given by the following gluing pattern . Given an edge $(g,gh_{j})$ of the Cayley graph ${\rm Cay}(G,H)$, the mark $gh_{j}\check{m}^{-j}_{i}$ on $S_{g}$ and the mark $gh_{j}m^{-j}_{i}$ on $S_{gh_{j}}$ are glued, for each $i\in\mathbb{N}$.

3. The proof of the Theorem 2

Let $G$ be a finitely generated subgroup of ${\rm GL}_{+}(2,\mathbb{R})$ without contracting elements and let $H$ be a finite generating set of $G$. The set $H$ can be written as $H=\{h_{j}:j\leq J\}$, for any positive integer $J$. We shall obtain the surface $S$ using the PSV construction. Then we shall prove that $S$ is a tame translation surface with Veech group $G$. Finally, we will describe the topological type of $S$.

3.1. Using the PSV construction to obtain the surface $S$

Step 1. The decorated surface.

For each $j\leq J$, we consider $\mathbb{E}(j,1)$ and $\mathbb{E}(j,2)$ copies of the Euclidean plane, which are endowed with a fixed origin 0 and an orthogonal basis $\beta=\{e_{1},e_{2}\}$. We shall define marks on these surfaces and they will be described by their ends points. On $\mathbb{E}(j,1)$ we draw the families of marks

[TABLE]

On $\mathbb{E}(j,2)$ we take the family of marks

[TABLE]

and the mark given by

[TABLE]

Finally, we glue the mark $l_{i}\in L\subset\mathbb{E}(j,1)$ to the mark $l^{{}^{\prime}}_{i}\in L^{{}^{\prime}}\subset\mathbb{E}(j,2)$ and thus, we obtain a tame Loch Ness monster

[TABLE]

which is called the buffer surface associated to the element $h_{j}$ of $H$ (see Figure 2).

Remark 2**.**

The buffer surface $S(Id,h_{j})$ is a modification of the surface appearing in the Construction 4.4 in [PSV].

We emphasize that the family of marks $\check{M}^{j}$ and the mark $h_{j}\check{m}^{-j}$ drawn on $S(Id,h_{j})$ have not yet been glued. In addition, the set of singular points of $S(Id,h_{j})$ is conformed by an infinitely many cone angle singularities of angle $4\pi$.

On the other hand, we take $\mathbb{E}$ the Euclidean plane endowed with a fixed origin $\overline{\textbf{0}}$ and an orthogonal basis $\beta=\{e_{1},e_{2}\}$. Analogously, We shall draw marks on this surface and they will be given by their ends points. For each $j\leq J$, on $\mathbb{E}$ we define the families of marks

[TABLE]

Now, we shall recursively draw new marks on $\mathbb{E}$. For $j=1$, we choose two suitable real numbers $x_{1}>0$ and $y_{1}<0$, and we define the mark

[TABLE]

on $\mathbb{E}$ such that it is disjoint from the families of marks $M$ and $M^{j}$, for each $j\in J$.

For $n\leq J$, we choose two suitable real numbers $x_{n}>0$ and $y_{n}<0$, and we define the mark

[TABLE]

on $\mathbb{E}$ such that it is disjoint from the families of marks $M$ and $M^{j}$, for each $j\in J$. Moreover, the mark $m^{-n}$ is also disjoint from the marks $m^{-1},\ldots,m^{-(n-1)}$ defined in the previous steps.

Let $\pi:\tilde{\mathbb{E}}\to\mathbb{E}$ be the three fold cyclic covering of $\mathbb{E}$ branched over the origin. Then we denote as $\tilde{M}:=\{\tilde{m_{i}}:\forall i\in\mathbb{N}\}$ one of the three sets of marks on $\tilde{\mathbb{E}}$ defined by $\pi^{-1}(M)$. Now, we take on $\mathbb{E}$ the marks $t_{1}:=(e_{2},\,2e_{2})$, and $t_{2}:=(-e_{2},\,-2e_{2})$, the purpose of these marks is to obtain new marks on $\tilde{\mathbb{E}}$. Then we denote as $\tilde{t_{1}}$ and $\tilde{t_{2}}$ one of the three marks on $\tilde{\mathbb{E}}$ defined by $\pi^{-1}(t_{1})$ and $\pi^{-1}(t_{2})$, respectively, such that they are on the same fold of $\tilde{\mathbb{E}}$ as $\tilde{M}$.

Finally, we take the union $\mathbb{E}\cup\tilde{\mathbb{E}}\cup_{j\in J}S(Id,h_{j})$ (see equation ( 2)) and glue marks as follows:

The marks $\tilde{t_{1}}$ and $\tilde{t_{2}}$ on $\tilde{\mathbb{E}}$ are glued. 2. 2.

The mark $m_{i}\in\mathbb{E}$ is glued to the mark $\tilde{m_{i}}\in\tilde{\mathbb{E}}$, for each $i\in\mathbb{N}$. 3. 3.

The mark mark $m_{i}^{j}\in\mathbb{E}$ is glued to the mark $\check{m}_{i}^{j}\in S(Id,h_{j})$, for each $i\in\mathbb{N}$ and each $j\in J$.

Thus, we obtain the tame Loch Ness monster

[TABLE]

which is called the decorated surface (see Figure 3).

Remark 3**.**

The surface $S_{{\rm dec}}$ is a slight alteration of the surface appearing in the Construction 4.6 in [PSV], where the authors introduced a tame Loch Ness monster with infinitely many marks on it, we consider the same surface but with only some of these marks.

Remark 4**.**

We highlight that for each $j\leq J$, the marks $h_{j}\check{m}^{-j}$ and $m^{-j}$ on the decorated surface $S_{{\rm dec}}$ have not yet been glued. Moreover, such surface $S_{{\rm dec}}$ has the following properties:

Its set of singular points is conformed by an infinitely many cone angle singularities of angle $4\pi$ and, only one cone angle singularity of angle $6\pi$, which is denoted by $\tilde{\textbf{0}}$. 2. 2.

There are only three saddle connections $\gamma_{1}$, $\gamma_{2}$, $\gamma_{3}$ such that each one of them have as one of their end points the singularity $\tilde{\textbf{0}}$ (see Figure 3). The holonomy vectors of these saddle connections are $\{\pm e_{1},\pm e_{2}\}$.

Step 2. The puzzle associated to the triplet $(1,G,H)$.

Let $S_{g}$ be the affine copy of the decorated surface $S_{{\rm dec}}$, for each element $g\in G$. We denote by $gh_{j}\check{m}^{-j}$ and $gm^{-j}$ (respectively) the marks on $S_{g}$, which are given by the image of the marks $h_{j}\check{m}^{-j}$ and $m^{-j}$ (respectively) via the affine diffeomorphism $f_{g}:S_{{\rm dec}}\to S_{g}$, where $j\leq J$. Thus, we obtain the puzzle associated to the triplet $(1,G,H)$ given by $\mathfrak{P}(1,G,H):=\{S_{g}:g\in G\}$. The following lemma will be use to prove the tameness of our surface $S$.

Lemma 3.1** (Lemma 4.5 [PSV]).**

For every $g\in G$, the distance in $S_{g}$ between the families of marks $\{gh_{j}\check{m}^{-j}:j\leq J\}$ and $\{gm^{-j}:j\leq J\}$ is at least $1/\sqrt{2}$.

Step 3. The assembled surface $S$ to the puzzle $\mathfrak{P}(1,G,H)$.

We consider the union $\cup_{g\in G}S_{g}$ and glue marks as fallows.

$\ast$ Given the edge $(g,gh_{j})$ of the Cayley graph $Cay(G,H)$, then we glue the mark $gh_{j}\check{m}^{-j}$ on $S_{g}$ to the mark $gh_{j}m^{-j}$ on $S_{gh_{j}}$.

Let us observe that by construction the marks $gh_{j}\check{m}^{-j}$ and $gh_{j}m^{-j}$ are parallel, therefore, the gluing is well defined. Then the assembled surface to the puzzle $\mathfrak{P}(1,G,H)$ obtained under the above gluing is a translation surface, which will be denoted by $S$.

3.2. The surface $S$ is a tame translate surface and its Veech group is the subgroup $G<{\rm GL}_{+}(2,\mathbb{R})$

One can use several of the ideas described in [RamVal, Theorem 3.7] to easily prove these properties.

Tameness.

We must show that $S$ is a complete metric space with respect to its natural flat metric $d$ and, its set of singularities is discrete in $S$. Let $(\hat{S},\hat{d})$ be the metric completion space of $(S,d)$. For each $g\in G$, we consider the connected open subset

[TABLE]

which is obtained from $S_{g}$ (see equation (3)) by removing the marks $gh_{j}\check{m}^{-j}$ and $gm^{-j}$, for each $j\leq J$. Using the inclusion map, $S_{g}^{{}^{\prime}}$ it can be thought as a connected open subset of $S$. Then, $\overline{S_{g}^{{}^{\prime}}}$ the closure of $S_{g}^{{}^{\prime}}$ in $S$ is complete. If we take a Cauchy sequence $(x_{n})_{n\in\mathbb{N}}$ in $S$ and the real number $\varepsilon=\frac{1}{2\sqrt{2}}$, then there is a positive integer $N(\varepsilon)\in\mathbb{N}$, such that for all natural numbers $m,n\geq N(\varepsilon)$ the terms $x_{m},x_{n}$ satisfy $\hat{d}(x_{m},x_{n})<\varepsilon$. By Lemma 3.1, there is an element $g\in G$ such that the open ball $B_{\varepsilon}(x_{N(\varepsilon)})$ is contained in $\overline{S_{g}^{{}^{\prime}}}$. As $\overline{B_{\varepsilon}(x_{N(\varepsilon)})}\subset\overline{S(g)^{{}^{\prime}}}$ is complete, then the Cauchy sequence $(x_{n})_{n\in\mathbb{N}}$ is convergent in $\overline{B_{\varepsilon}(x_{N(\varepsilon)})}$. The discreteness is immediately from the Lemma 3.1.

The Veech group of $S$ is $G$.

Given that the group $G$ acts on $\mathfrak{P}(1,G,H):=\{S_{g}:g\in G\}$ by post composition on charts, then if we fix a matrix $\tilde{g}\in G$, for each element $g$ in $G$, there exists a natural affine diffeomorphism $f_{\tilde{g}g}:S_{g}\to S_{\tilde{g}g}$, satisfying the following properties:

The differential of $f_{\tilde{g}g}$ is the matrix $\tilde{g}$. 2. 2.

The map $f_{\tilde{g}g}$ send parallel marks to parallel marks.

Hence, the map $f:\bigcup\limits_{g\in G}S_{g}\to\bigcup\limits_{g\in G}S_{\tilde{g}g}$ defined by $f|_{S_{g}}:=f_{\tilde{g}g}$, is a gluing marks-preserving map. Then we obtain an affine diffeomorphism in the quotient $F_{\tilde{g}}:S\to S$ with differential matrix $\tilde{g}$. Thus, we conclude that $G<\Gamma(S)$. Conversely, we consider $f:S\to S$ an affine orientation preserving diffeomorphism different to the identity. From Remark 4, for each element $g\in G$, the surface $S$ has one singularity of angle $6\pi$, which is denoted by $\tilde{\textbf{0}}_{g}$. There are only three saddle connections $\gamma_{1}^{g}$, $\gamma_{2}^{g}$, $\gamma_{3}^{g}$ such that each one of them have that singularity as one of their end points. The holonomy vectors associated to these saddle connections are $\{\pm g\cdot e_{1},\pm g\cdot e_{2}\}$. Then the function $f$ sends the singularity $\tilde{\textbf{0}}_{{\rm Id}}$ to the singularity $\tilde{\textbf{0}}_{g}$, for some $g\in G$ and, $df$ the differential matrix of $f$ must map $\{\pm e_{1},\pm e_{2}\}$ to $\{\pm g\cdot e_{1},\pm g\cdot e_{2}\}$. The only possibility is $df=g$. Hereby, we conclude that $\Gamma(S)<G$.

3.3. Description of the topological type of $S$

We shall prove the following two assertion:

If $G$ is finite, then the surface $S$ has ${\rm card}(G)$ ends and each one of them have infinite genus. 2. 2.

Otherwise, if $G$ is not finite, then the topological type of $S$ can be represented in the form

[TABLE]

where $\mathcal{B}$ is a closed subset of ${\rm Ends}(S)$ homeomorphic to ${\rm Ends(G)}$ and, $\mathcal{U}$ is a countable dense open subset of ${\rm Ends}(S)$.

Case: The group $G$ is finite.

The group $G$ has cardinality ${\rm card}(G)=k$ for any natural number $k\in\mathbb{N}$. Let $K$ be a compact subset of $S$, we must prove that there exists a compact subset $C\subset S$ such that $K\subset C$ and $S\setminus C$ is conformed by $k$ open connected components and each one of them have infinite genus.

For each $g\in G$, the affine copy $S_{g}$ is homeomorphic to the Loch Ness monster (see equation (3). As the generating set $H$ of $G$ is finite, then the set of marks $\{gh_{j}\check{m}^{-j},gm^{-j}:j\leq J\}$ on $S_{g}$ is finite. We consider the connected subsurface $S^{{}^{\prime}}_{g}$ of $S_{g}$ (see equation 4)). This subsurface $S^{{}^{\prime}}_{g}$ has infinite genus and via the inclusion map it can be thought as a connected subsurface of $S$ with infinite genus. In addition, the boundary $\partial S^{{}^{\prime}}_{g}$ of $S^{{}^{\prime}}_{g}$ in $S$ is compact, because it is conformed by a finitely many disjoint closed curves. As $G$ is finite, then the set

[TABLE]

is conformed by exactly $k$ open connected components, each one of them have infinite genus. Let $K_{g}$ be the closure of the set $K\cap S^{{}^{\prime}}_{g}$ in $S_{g}$, for each $g\in G$. As $C_{g}$ is a compact subset of $S_{g}$, then there exists a compact subset $C_{g}\subset S_{g}$ such that $K_{g}\cap\{gh_{j}\check{m}^{-j},gm^{-j}:j\leq J\}\subset C_{g}$ and $S_{g}\setminus C_{g}$ is conformed by an open connected with infinite genus. We take $C$ the closure of $\bigcup_{g\in G}(C_{g}\setminus\{gh_{j}\check{m}^{-j},gm^{-j}:j\leq J\})$ in $S$. As $G$ is finite, then $C$ is a compact subset of $S$ and by the construction of $C$ we hold that $K\subset C$ and the set

[TABLE]

is conformed by $k$ open connected components, each one of them have infinite genus.

Case: The group $G$ is not finite.

The sketch of the proof is the following. We shall define the set $\mathcal{U}$ from the ends of the affine copies $S_{g}$ and will prove that it is a countable discrete open subset of ${\rm Ends}(S)$. Then, we shall give an appropriate embedding $i_{\ast}$ from ${\rm Ends}(G)$ to ${\rm Ends}(S)$ and obtain the set $\mathcal{B}$. Using an embedding from the Cayley graph ${\rm Cay}(G,H)$ to the surface $S$ we shall prove the equality ${\rm Ends}(S)={\rm Ends}_{\infty}(S)=\mathcal{B}\sqcup\mathcal{U}$. Finally, we shall show that $\mathcal{U}$ is a dense subset of ${\rm Ends}(S)$.

Step 1.

For each $g\in G$, we take the subsurface $S^{{}^{\prime}}_{g}\subset S_{g}$ defined in the equation (4). Let $[U(g)_{n}]_{n\in\mathbb{N}}$ be the only one end of the Loch Ness monster $S_{g}$. We can suppose without of generality that $U(g)_{n}\subset S^{{}^{\prime}}_{g}$ for each $n\in\mathbb{N}$. Via the inclusion map, the surface $S^{{}^{\prime}}_{g}$ can be thought as a subsurface of $S$, then the sequence $(U(g)_{n})_{n\in\mathbb{N}}$ of $S_{g}$ defines an end of infinite genus in the surface $S$.

Remark 5**.**

For each $g\in G$, the boundary $\partial S^{{}^{\prime}}_{g}$ of the subsurface $S_{g}\subset S$ is compact, because it is conformed by a finitely many disjoint closed curves. In addition, for any two elements $g\neq\tilde{g}\in G$, the subsurfaces $S^{{}^{\prime}}_{g}$ and $S^{{}^{\prime}}_{\tilde{g}}$ are disjoint.

From the previous remark, we obtain the countable set $\mathcal{U}$ conformed by different ends of $S$, given by

[TABLE]

Let us remark that the subset $\mathcal{U}\subset{\rm Ends}(S)$ is discrete, because for every $g\in G$ the open subset $U(g)_{1}$ of $S$ with compact boundary $\partial U(g)_{1}$ in $S$, defines the open subset $(U(g)_{1})^{\ast}$ of ${\rm Ends}(S)$, which satisfies $(U(g)_{1})^{\ast}\cap\mathcal{U}=\{[U(g)_{n}]_{n\in\mathbb{N}}\}$.

Step 2.

We shall define the embedding $i_{\ast}:{\rm Ends}(G)\hookrightarrow{\rm Ends}(S)$. Let $\overline{S^{{}^{\prime}}}_{g}$ be the closure in $S$ of the surface $S^{{}^{\prime}}_{g}$ (see the equation (4)). Given $W$ a non-empty connected open subset of ${\rm Cay}(G,H)$ with compact boundary $\partial W$, we can suppose without loss of generality that the boundary $\partial W\subset V({\rm Cay}(G,H))=G$, then we define the subset $\tilde{W}\subset S$ given by

[TABLE]

This set $\tilde{W}$ is a non-empty connected open subset of $S$ with compact boundary. Moreover, it is a subsurface with infinite genus. In the following Remark we state two properties of this object, which can be easily deduced.

Remark 6**.**

Given $W$ and $V$ two non-empty open connected subsets of ${\rm Cay}(G,H)$ each one them have compact boundary $\partial W$ and $\partial V$, respectively, such that $\partial W,\partial V\subset G$, then

$\ast$* If $W\supset V$, then $\tilde{W}\supset\tilde{V}$.*

$\ast$* If $W\cap V=\emptyset$, then $\tilde{W}\cap\tilde{V}=\emptyset$.*

From remark above, the end $\left[W_{n}\right]_{n\in\mathbb{N}}$ of the group $G$ naturally defines the end with infinite genus $[\tilde{W}_{n}]_{n\in\mathbb{N}}$ of the surface $S$. Hence, we obtain a well define map $i_{\ast}:{\rm Ends}(G)\to{\rm Ends}(S)$, given by

[TABLE]

We claim that $i_{\ast}$ is an embedding. We must show that it is injective. Let $[W_{n}]_{n\in\mathbb{N}}$ and $[V_{n}]_{n\in\mathbb{N}}$ be two different elements in ${\rm Ends}(G)$, then there is a natural number $n\in\mathbb{N}$ such that $W_{n}\cap V_{n}=\emptyset$. Remark 6 implies $\tilde{W}_{n}\cap\tilde{V}_{n}=\emptyset$, therefore the ends $i_{\ast}([W_{n}]_{n\in\mathbb{N}})=[\tilde{W}_{n}]_{n\in\mathbb{N}}$ and $i_{\ast}([V_{n}]_{n\in\mathbb{N}})=[\tilde{V}_{n}]_{n\in\mathbb{N}}$ in ${\rm Ends}(S)$ are different.

To prove the continuity of $i_{\ast}$, we consider $[W_{n}]_{n\in\mathbb{N}}$ an end in ${\rm Ends}(G)$ and an open subset $V\subset S$ with compact boundary, such that $i_{\ast}([W_{n}]_{n\in\mathbb{N}})=[\tilde{W}_{n}]_{n\in\mathbb{N}}\in V^{\ast}\subset{\rm Ends}(S)$, it means that $\tilde{W}_{j}\subset V$, for some $j\in\mathbb{N}$. We must prove that there is a neighborhood $Z^{\ast}\subset{\rm Ends}(G)$ of $[W_{n}]_{n\in\mathbb{N}}$, such that $i_{\ast}\left(Z^{\ast}\right)\subset V^{\ast}$. By Remark 6, the choice of $Z^{\ast}=(W_{j})^{\ast}$ implies $i_{\ast}\left(Z^{\ast}\right)\subset V^{\ast}$.

Finally, the map $i_{\ast}$ is closed because every continuous map from a compact space to a Hausdorff space is closed. Hence, the map $i_{\ast}$ is an embedding.

We shall denote by $\mathcal{B}$ to the image $i_{\ast}({\rm Ends}(G))$. From the definition of the set $\mathcal{U}$ given in the equation (5), we conclude that $\mathcal{B}\cap\mathcal{U}=\emptyset$ and $\mathcal{B}\sqcup\mathcal{U}\subset{\rm Ends}(S)$.

Step 3.

Now, we will proceed to define an embedding $i:{\rm Cay}(G,H)\hookrightarrow S$. We shall describe the image of each vertex and each edge of ${\rm Cay}(G,H)$ under the map $i$.

For each element $g\in G$, we denote by $\overline{\textbf{0}}_{g}$ the point in the affine copy $S_{g}$, which is given by the image of the point $\overline{\textbf{0}}$ (see equation (3)) in the decorated surface $S_{{\rm dec}}$ via the affine diffeomorphism $f_{g}:S_{{\rm dec}}\to S_{g}$. Then the surface $S^{{}^{\prime}}_{g}$ described in the equation (4) has the point denoted by $\overline{\textbf{0}}_{g}$. Thus, we define the map $h:V({\rm Cay}(G,H))=G\to S$ given by

[TABLE]

On the other hand, for each $j\leq J$, there is a curve $\beta_{j}:[0,1]\to S$ satisfying the following properties:

$\ast$ The initial and terminal points of $\beta_{j}$ are $\overline{\textbf{0}}_{\rm Id}$ and $\overline{\textbf{0}}_{h_{j}}$, respectively. See Figure 4.

$\ast$ For each $i\neq j\leq J$, the intersection $\beta_{i}([0,1])\cap\beta_{j}([0,1])=\{\overline{\textbf{0}}_{\rm Id}\}$.

Given that the edge $({\rm Id},h_{j})$ of the Cayley graph ${\rm Cay}(G,H)$ is homeomorphic to the open interval $(0,1)$, then we can suppose without loss of generality that the curve $\beta_{j}$ is defined from $[Id,h_{j}]$ to $S$ such that $\beta_{j}({\rm Id})=\overline{\textbf{0}}_{\rm Id}$ and $\beta_{j}(h_{j})=\overline{\textbf{0}}_{h_{j}}$. As the Veech group of the surface $S$ is $G$, then for each element $g\in G$, there is an affine diffeomorphims $f_{g}:S\to S$ such that its differential is $df_{g}=g$. Thus, we get the composition curve

[TABLE]

satisfying the following properties:

$\ast$ The initial and terminal points of $f_{g}\circ\beta_{j}$ are $\overline{\textbf{0}}_{g}$ and $\overline{\textbf{0}}_{gh_{j}}$, respectively.

$\ast$ For each $i\neq j\leq J$, the intersection $f_{g}\circ\beta_{i}([0,1])\cap f_{g}\circ\beta_{j}([0,1])=\{\overline{\textbf{0}}_{g}\}$.

Similarly, given that the edge $(g,gh_{j})$ of the Cayley graph ${\rm Cay}(G,H)$ is homeomorphic to the open interval $(0,1)$, then we can suppose without loss of generality that the composition curve $f_{g}\circ\beta_{j}$ is defined from $[g,gh_{j}]$ to $S$ such that $f_{g}\circ\beta_{j}(g)=\overline{\textbf{0}}_{g}$ and $f_{g}\circ\beta_{j}(gh_{j})=\overline{\textbf{0}}_{gh_{j}}$.

From the equations (7) and (8) we get the embedding

[TABLE]

such that $i_{|G}:=h$ and $i_{|[g,gh_{j}]}:=f_{g}\circ\beta_{j}$, for each $g\in G$ and $j\leq J$.

Step 4.

We shall show the equality ${\rm Ends}(S)=\mathcal{B}\sqcup\mathcal{U}$. We must just prove that ${\rm Ends}(S)\subset\mathcal{B}\sqcup\mathcal{U}$. Let $[U_{n}]_{n\in\mathbb{N}}$ be an end of $S$. As $S=\bigcup\limits_{g\in G}\overline{S^{{}^{\prime}}_{g}}$, then for each $n\in\mathbb{N}$, we consider the subset $G(n)=\{g\in G:\overline{S^{{}^{\prime}}_{g}}\cap U_{n}\}\subset G$ and, define the open subset

[TABLE]

Given $U_{n}$ is an non-empty open connected subset of $S$ with compact boundary such that $U_{n}\supset U_{n+1}$, then $Z_{n}$ is also an open connected subset of $S$ with compact boundary such that $Z_{n}\supset Z_{n+1}$, for each $n\in\mathbb{N}$. Using the definition of end and the construction of $Z_{n}$, it is easy to show that the sequences $(Z_{n})_{n\in\mathbb{N}}$ and $(U_{n})_{n\in\mathbb{N}}$ defines the same end of $S$, in other words, $[U_{n}]_{n\in\mathbb{N}}=[Z_{n}]_{n\in\mathbb{N}}$. Then, one of the following situations must be happen:

$\ast$ There is an element $g\in G$ and natural number $M(g)\in\mathbb{N}$, such that for all $m\geq M(g)$ it satisfies $Z_{m}\subset S^{{}^{\prime}}_{g}$. This implies that the sequences $(Z_{n})_{n\in\mathbb{N}}$ and $(U(g)_{n})_{n\in\mathbb{N}}$ must be equivalent (see equation (5)). It proves that $[U_{n}]_{n\in\mathbb{N}}\in\mathcal{U}$.

$\ast$ Otherwise, for all $n\in\mathbb{N}$, the subset $G_{n}\subset G$ is infinite. As the embedding $i$ described in (9) is a continuous map, then the inverse image ${\hat{Z}}_{n}:=i^{-1}(Z_{n}\cap i({\rm Cay}(G,H)))$ is an open connected subset of ${\rm Cay}(G,H)$ with compact boundary, for each $n\in\mathbb{N}$. Moreover, the sequence $(\hat{Z}_{n})_{n\in\mathbb{N}}$ defines an end of the group $G$. By the construction of the sequence $(Z_{n})_{n\in\mathbb{N}}$ of $S$, we hold that the embedding $i_{\ast}$ defined in (6) sends the end $[\hat{Z}_{n}]_{n\in\mathbb{N}}\in{\rm Ends}(G)$ to the end $[Z_{n}]_{n\in\mathbb{N}}\in{\rm Ends}(S)$. Therefore, $[Z_{n}]_{n\in\mathbb{N}}$ have to be in $\mathcal{B}$. Thus, we conclude that ${\rm Ends}(S)=\mathcal{B}\sqcup\mathcal{U}$.

Step 5.

As $\mathcal{U}$ is an open subset of ${\rm Ends}(S)$ (see equation (5)), then its complement ${\rm Ends}(S)\setminus\mathcal{U}=\mathcal{B}$ is a closed subset of ${\rm Ends}(S)$. We shall prove that $\mathcal{U}$ is dense. Let $[Z_{n}]_{n\in\mathbb{N}}$ be an end of $\mathcal{B}$, we must show that this ends belongs to the closure of $\mathcal{U}$. We consider a non-empty connected open $U$ of $S$ with compact boundary, such that the open subset $U^{\ast}\in{\rm Ends}(S)$ contains the element $[Z_{n}]_{n\in\mathbb{N}}$. Then there exists an element $\tilde{g}\in\{g\in G:\overline{S^{{}^{\prime}}_{g}}\cap U\}$, such that the subsurface $S_{\tilde{g}}^{{}^{\prime}}\subset U$. This condition implies that the end $[U(\tilde{g})_{n}]_{n\in\mathbb{N}}$ of $\mathcal{U}$ belongs to $U^{\ast}$. Therefore, the end $[Z_{n}]_{n\in\mathbb{B}}$ is in the closure of $\mathcal{U}$.

∎

Acknowledgements

The author was partially supported by UNIVERSIDAD NACIONAL DE COLOMBIA, SEDE MANIZALES, Project Hermes 47019. Camilo Ramírez Maluendas has dedicated this work to his beautiful family: Marbella and Emilio, in appreciation of their love and support.