On Infinitely generated Fuchsian groups of the Loch Ness monster, the Cantor tree and the Blooming Cantor tree

Abstract In this paper, for a non-compact Riemman surface S homeomorphic to either: the Infinite Loch Ness monster, the Cantor tree and the Blooming Cantor tree, we give a precise description of an infinite set of generators of a Fuchsian group Γ < PSL(2, ℝ), such that the quotient space ℍ/Γ is a hyperbolic Riemann surface homeomorphic to S. For each one of these constructions, we exhibit a hyperbolic polygon with an infinite number of sides and give a collection of Mobius transformations identifying the sides in pairs.


Introduction
A classical problem during the 19th century, in which several authors had been focused e.g., Klein, Schwarz, and Poincaré among others, known as the uniformization problem, [1], said that: being S a Riemann surface, nd all domains S ⊂Ĉ and holomorphic functions t : S → S such that at each point p ∈ S, t is a local uniformizing variable at p. Equivalently, from the point of view of the Covering Spaces theory, there is a topological disc B ⊂ S with center p such that the restriction of t to each component of t − (B) is a homeomorphism. It will means that the triplet ( S, S, t) is a covering space with base space S, total space S, and holomorphic projection t. For S being compact, it holds the rst version of the Uniformization Theorem due to Klein.
a. The Cantor tree.
b. The Blooming Cantor tree. The paper is organized as follows: In section 2 we collect the principal tools used through the paper such as geometrical construction of the Cantor set, the classi cation theorem of non-compact and orientable surfaces, and a short introduction to the hyperbolic plane and Geometric Schottky groups. Finally, section 3 is dedicated to the proof of our main results.

Preliminaries . Geometrical construction of the Cantor set
For the Cantor set we begin its geometrical construction by removing the middle third started with the closed interval I := [ , ] ⊂ R. We denotes I = , + ∪ + , the closed subset of I held from I by removing its middle third + , + . The closed subset I ⊂ I is the union of two disjoint closed intervals having length . We denotes I = , + ∪ + , + ∪ + , + ∪ + , be the closed subset of I held from I by removing its middle thirds + , + and + , + respectively. The closed subset I ⊂ I is the union of four disjoint closed intervals having length . We now construct inductively the closed subset In ⊂ I from I n− by removing its middle thirds. On the other hand, we note that each positive integer number k can be written in the binary form k = t · + t · + . . . + t i · i + . . . + tm · m , where tm = , t i ∈ { , } for all i ∈ { , . . . , m} and any m ∈ N. Thus, we de ne s k as following s k = · t · + · t · + . . .
Contrary, if k = then s k = . Therefore, the intersection of closed subset ω = ∩ n∈N In is well-known as the Cantor set, which is the only totally disconnected, perfect compact metric space (up to homeomorphism, see [22,Corollary 30.4]).

. Ends spaces
De nition 2.3. [9]. Let X be a locally compact, locally connected, connected Hausdor space, and let U ⊃ U ⊃ · · · be an in nite nested sequence of non-empty connected open subsets of X, so that, the boundary of Un in X is compact for every n ∈ N, n∈N U = ∅, and for any compact subset K of X there is l ∈ N such that U l ∩ K = ∅. We shall denote the sequence U ⊃ U ⊃ · · · as (Un) n∈N . Two sequences (Un) n∈N and (U n ) n∈N are equivalent if for any l ∈ N there is k ∈ N such that U l ⊃ U k and, similarly, for any n ∈ N it exists m ∈ N such that U n ⊃ Um. The corresponding equivalence classes are called the topological ends of X. We will denote the space of ends by Ends(X) and each equivalence class [Un] n∈N ∈ Ends(X) is called an end of X. n∈N is called planar if there is l ∈ N such that U l is planar. The genus of a surface S is the maximum of the genera of its compact subsurfaces. Remark that, if a surface S has in nite genus, there is no nite set C of mutually non-intersecting simple closed curves with the property that S \ C is connected and planar. We de ne Ends∞(S) ⊂ Ends(S) as the set of all ends of S which are not planar (ends with in nity genus). It comes from the de nition that Ends∞(S) forms a closed subspace of Ends(S). Theorem 2.5 (Classi cation of non-compact and orientable surfaces, [13], [19] Of all non-compact surfaces, our interest points at three of them. The rst one is that surface which has in nite genus and only one end. It is called the Loch Ness monster (see Figure 1). Remark that a surface S has only one end if and only if for all compact subset K ⊂ S there is a compact K ⊂ S such as K ⊂ K and S \ K is connected (see [21]). The other two remaining surfaces are the Cantor tree and the Blooming Cantor tree (see Figure 2), which have ends space homeomorphic to the Cantor set, although in the rst one all ends are planar, while the ends of the second one are all not planar (see [10]).
Cantor binary tree. For every n ∈ N let n = n i= { , } i and let π i : n → { , } be the projection onto the i-th coordinate. We de ne V = {Ds : Ds ∈ n for some n ∈ N} and E as the union of (( ), ( )) with the set {(Ds , D t ) : Ds ∈ n and D t ∈ n+ for some n ∈ N, and π i (Ds) = π i (D t ) for every i ∈ { , ..., n}}. The in nite 3-regular tree with vertex set V and edge set E will be called the Cantor binary tree and denoted by T ω , (see Figure 3).

. Hyperbolic plane
The upper half-plane H = {z ∈ C : Im(z) > } equipped with the riemannian metric ds = Let C be a half-circle whose end points are in R and whose center is α ∈ R. There is a one-real-dimensional familiy of order two Möbius transformations in PSL( , R) permuting these two end points. Each of them is uniquely determined by choosing a point on C as its xed point. In particular, there is exactly one of such a conformal involutions f C whose xed point has real part α, called the inversion in C. If r > is the radius of C, then f C is given as follows The setČ(f ) = {z ∈ H : |z − α| < r} is called the inside of C. Contrary, the setĈ(f ) := {z ∈ H : |z − α| > r} is called the outside of C.
Remark 2.10. We let L α− r and L α+ r be the two orthogonal straight lines to the real axis R through the points α − r and α + r, respectively. Then the inversion f C sends L α− r (analogously, L α+ r ) onto the half-circle whose ends points are α + r and α (respectively, α − r and α), (see Figure 4). Given that f C is an element of PSL( , R), then for every ϵ < r the closed hyperbolic ϵ-neighborhood of the half-circle C does not intersect any of the hyperbolic geodesics L α− r , f C (L α− r ), L α+ r , and f C (L α+ r ). Lemma 2.11. Let C and C be two disjoint half-circles having centers and radius α , α ∈ R, and r , r > , respectively. Suppose that |α −α | > (r +r ), then for every ϵ < min{r ,r } the closed hyperbolic ϵ-neighborhoods of the half-circles C and C are disjoint.
Proof. By hypothesis |α − α | > (r + r ), then the open strips S = {z ∈ H : α − r < Re(z) < α + r } and S = {z ∈ H : α − r < Re(z) < α + r } are disjoint. We remark that the half-circle C i belongs to the open strip S i , for every i ∈ { , }. Further, f C the inversion in C, sends the open strip S onto the open strip S and vice versa. We denote r = min{r , r }. For each ϵ < r , let If Γ is a Fuchsian group such that none of its non-trivial elements xes ∞ (i.e., c ≠ ), then the subset R of H de ned as follows

. Fuchsian groups and Fundamental region
is a fundamental domain for the group Γ (see e.g., [8], [15,Theorem H.3], [11,Theorem 3.3.5]). The fundamental domain R is well-known as the Ford region for Γ. On the other hand, we can get a Riemann surface from any Fuchsian group Γ. It is only necessary to de ne the action α : , which is proper and discontinuous (see [12,Theorem 8. 6]). Now, we de ne the subset We note that the subset K is countable and discrete. Moreover, the action α leaves invariant the subset K. Then the action α restricted to the hyperbolic plane H removing the subset K is free, proper and discontinuous. Therefore, the quotient space S = (H − K)/Γ is well-de ned and via π : . It comes with a hyperbolic structure, it means, S is a Riemann surface (see e.g., [14], [20,Theorem 18.2]).

. Geometric Schottky groups
Consider an open interval A in the real line R. Let ∂A denote its boundary in R, which consists of two points. The two points of ∂A determine a geodesic γ A in H. This geodesic in turn de nes two closed half-planes in H. The boundary at in nity of one of these two closed half-planes is equal to the closure A = A ∪ ∂A and we will denote this closed half-plane byÂ. In Figure 5 are depicted A, γ A andÂ. A subset I ⊆ Z is called symmetric if it satis es that ∉ I and for every k ∈ I implies −k ∈ I. We note that any Schottky description U(A k , f k , I) de nes a Geometric Schottky group.

Proposition 2.17. [23, Proposition 4] Every Geometric Schottky group Γ is a Fuchsian group.
The standard fundamental domain for the Geometric Schottky group Γ having Schottky description

Proposition 2.18. [23, Proposition 2] The standard fundamental domain F(Γ) is a fundamental domain for
the Geometric Schottky group Γ.

Main result
The proof of our main results are based on the following sketch. First, we shall build explicitly a suitable family of mutually disjoint half circles C. Then we shall de ne the set J composed by the Möbius transformations having as isometric circles the elements of C. Immediately, we will prove that Γ the subgroup of PSL( , R) generated by J is a Fuchsian group. Also, we show that the quotient space H/Γ is the desired non-compact surface.
. Proof Theorem 1.3 Step 1. Building the group Γ. Given C the family composed by the half-circle with centers the even integers on the real line R and radius one (see Figure 6), we consider Γ the Fuchsian subgroup of PSL( , Z), which is generated by the set {fn , gn : n ∈ Z}, where For each n ∈ Z, the Möbius transformations fn and f − n have as isometric circles the half-circles C(fn) and C(f − n ) of C, respectively, whose centers are n and n + , respectively. Analogously, the Möbius transformations gn(z) and g − n (z) have as isometric circles the half-circles C(gn) and C(g − n ) of C, respectively, whose centers are n + and n + , respectively.
On the other hand, Γ acts freely and properly discontinuously on H, because the intersection of any two di erent elements belonged to C is either: empty or at in nity, it means, they meet at the same point in the real line R. Then the quotient space S = H/Γ is a well-de ned hyperbolic Riemann surface via the projection map π : H → S.
Step 2. The desired surface. To end the proof we must prove that S is the In nite Loch Ness monster, i.e., it has in nite genus and only one end. The Ford region R associated to Γ (see Figure 7) is a connected and locally nite subset of H, having in nite hyperbolic area. Further, its boundary is C, a family conformed by half-circles mutually disjoint. Given that R the fundamental domain of Γ is a noncompact Dirichlet region having in nite hyperbolic area, then the quotient space S is also a non-compact hyperbolic surface with in nite hyperbolic area (see [12,Theorem 14.3]). The surface S has only one end. Let K be a compact subset of S. We must prove that there is a compact subset K ⊂ K ⊂ S such that S \ K is connected. Given that the quotient R /Γ is homeomorphic to S we must suppose that there is a compact subset B ⊂ R such that the projection map π sends B to K i.e., π(B) = K. The hyperbolic plane H has exactly one end, then there exist two closed intervals I , I ⊂ R such that B ⊂ I × I ⊂ H, and the di erence H \ (I × I ) is connected. Then π((I × I ) ∩ R ) = K is a compact subset of S such that K ⊆ K . We claim that S \ K is connected. Let [z] and [w] be two di erent points belonged to S \ K we shall build a path in S \ K joining both points.
The following set are necessary to the proof. For every x ∈ R \ I we denote as γx the perpendicular line to the real line, passing by x. For every y > with y ∉ I we consider the connected set γ y = {z ∈ H : Im(z) = y}. Then the intersection γx ∩ (I × I ) = ∅, and the projection map π sends the set γx ∩ R into a connected subset of S. The intersection γ y ∩ (I × I ) = ∅, and the projection map π sends the set γ y ∩ R into a connected subset of S.
From the equivalent classes [z] ≠ [w] ∈ S \ K we can assume without loss of generality that z, w ∈ R \ (I × I ), then there exist two connected subsets γ and γ such that z ∈ γ and w ∈ γ . If γ ∩ γ ≠ ∅ then the image of (γ ∪ γ ) ∩ R under π is a connected subset belonged to S \ K containing the points [z] and [w]. Oppositely, if γ ∩ γ = ∅, then there is a connected subset γ * such that γ ∩ γ * ≠ ∅ and γ ∩ γ * ≠ ∅. Consequently, the image of (γ ∪ γ ∪ γ * ) ∩ R under π is a connected subset belonged to S \ K containing the points [z] and [w]. This proves that the subset S \ K is connected.
The surface S has in nite genus. For every n ∈ Z, we de ned the subset Sn = {z ∈ H : < Im(z) < and − + n < Re(z) < + n} ⊂ H.
The projection map π sends the intersection Sn ∩ R into a subsurface with boundary Sn ⊂ S, which is homeomorphic to the torus punctured by only one point (see Figure 8). Furthermore, for any two di erent integers m ≠ n ∈ Z the subsurfaces Sn and Sm are disjoint. Thus, we conclude that the hyperbolic surface S has in nite genus.
Let C(f , ) and C(f − , ) be the two half-circles having centers α , and α − , respectively, and the same radius r( ) = · , (see Figure 9). Now, we take the Möbius transformations Finally, we de ne the sets By construction, each Möbius transformations of J is hyperbolic and the half-circles of C are pairwise disjoint.
Hence, we let C(f n,k ), C(f − n,k ) be the two half-circles having centers α n,k and α − n,k correspondingly, and the same radius r(n) = n · (see Figure 10). Now, for each k ∈ { , . . . , n− − } we take the Möbius transformations which have as isometric circles C(f n,k ) and C(f − n,k ), respectively. Using Remark 2.9 we have c n,k = r(n) = n · , d n,k = −c n,k · α n,k = − ·( n · + + s k− ), a n,k = c n,k · α − n,k = − ·( n · + + s k− ). Now, we substitute these values in the equation a n,k · d n,k − c n,k · b n,k = and computing we hold b n,k = · ( n · + + s k− ) − Hence, we denote the subgroup of PSL( , R) generated by the set J by Γ. We observe that by construction each Möbius transformation of J is hyperbolic and the half-circles of C are pairwise disjoint.
Step 2. The group Γ is a Fuchsian group. In order to Γ will be a Geometric Schottky group, we shall de ne a Schottky description for it. Hence, by Proposition 2.17 we will conclude that Γ is Fuchsian. The elements belonged to the set J can be indexed by a symmetric subset of Z. Merely, we consider P = {pn} n∈N the subset of all the prime numbers. Then it is easy to check that the map ψ : J → Z de ned by is well-de ned and injective. The image of J under ψ is a symmetric subset of Z, which we denote as I. Given that for each element k belonged to I there is a unique transformation f ∈ J such that ψ(f ) = k, then we label the map f as f k and its respective isometric circle as C(f k ). Hence, we re-write the sets J and C as follows On the other hand, we de ne the set {A i } i∈I where A i is the straight segment in the real line R whose ends points coincide to the ends points at the in nite of the half-circle C(f i ) (see equation (10)). We claim that the pair ({A i∈I is a Schottky description. By the inductive construction of the set J := {f i } i∈I described above, it is immediate that the pair ({A i }, {f i }) i∈I satis es the conditions 1 and 2 of the De nition 2.14. Thus, we must only prove that the condition 3 of De nition 2.14 is done. For any i ∈ I, we denote as α i and r i the center and radius of C(f i ). Hence, we de ne the open strip Figure 4). We note that C(f i ) ⊂ S i and for any two di erent transformations f i ≠ f j ∈ J its respective open strips associated S i and S j are disjoint. Further, by construction of the family C (see equation (10)) there exist a transformation f k ∈ J such that r k , the radius of its isometric circle C(f k ) satis es r i r k for all i ∈ I. Moreover, by Remark 2.10 it follows that B k ϵ , the closed hyperbolic ϵ-neighborhood of the half-circle i∈I is a Schottky description. Step 3. Holding the surface called the Cantor tree. The Geometric Schottky group Γ acts freely and properly discontinuously on the hyperbolic plane H, because the intersection of any two di erent elements of C is empty. Then the quotient space S := H/Γ is well-de ned and through the projection map π : H → S de nes as z → [z] is a hyperbolic surface. To end the proof we shall show that S is the Cantor tree i.e., its ends space is homeomorphic to the Cantor set having all ends planar. To prove this, we will describe the ends space of S using the property of σ-compact of S and showing that there is a homeomorphism f from the ends spaces of the Cantor binary tree Ends(T ω ) onto the ends space Ends(S). it is a fundamental domain for Γ having the following properties. [12,Theorem 14.3]). We note that by construction, the surface S does not have genus i.e., its ends are planar.

It is connected and locally nite having in nite hyperbolic area. Further, its boundary is composed by the family of half-circle C (see equation (10)). In other words, it consists of in nitely many hyperbolic geodesic with ends points at in nite and mutually disjoint. 2. It is a non-compact Ford region and the quotient F(Γ)/Γ is homeomorphic to S, then the quotient space S is also a non-compact hyperbolic surface with in nite hyperbolic area (see
Since surfaces are σ-compact space, for S there is an exhaustion of S = n∈N Kn by compact sets whose complements de ne the ends spaces of the surface S. More precisely; For n = . We consider the radius r( ) = · given in the recursive construction of Γ and de ne the compact subsetK = {z ∈ H : − ≤ Re(z) ≤ , and r( ) ≤ Im(z) ≤ }. The image π(K ∩ F(Γ)) = K is a compact subset of S and, the di erence S \ K consists of two connected components whose closure in S is non-compact, but they have compact boundary. Hence, we can write S \ K = U U . The set of connected components of S \ K and the set = { , } are equipotent. For n. We consider the radius r(n) = n · given in the recursive construction of Γ and de ne the compact subsetKn = {z ∈ H : −(n + ) ≤ Re(z) ≤ n + , and r(n) ≤ Im(z) ≤ n}. By constructionK n− ⊂Kn and the image π(Kn ∩ F(Γ)) = Kn is a compact subset of S just as K n− ⊂ Kn. By de nition ofKn the di erence S \ Kn consists of n connected components whose closure in S is non-compact, but they have compact boundary. Moreover, for every l in the set n− there exist exactly two connected components of S \ K n− contained in  . Proof Theorem 1.5 Step 1. Building the group Γ.
For n = . Building the set J containing in nitely countable Möbius transformation and the set C composed by its respective isometric circles. We let C(f , ) and C(f − , ) be two half-circles having centers α , and α − , , respectively (as in equation (6)), and the same radius r( ) = · · (see Figure 11). Now, we consider the Möbius transformations which has as isometric circles C(f , ) and C(f − , ), respectively. By Remark 2.9 we have a , = − , c , = , d , = − . From the relation a , · d , − b , · c , = we hold b , = . By construction the Möbius transformations f , and f − , are hyperbolic and the half-circles C(f , ) and C(f − , ) are pairwise disjoint. Now, we shall build sequences of half-circles at the left and at the right of C(f , ), C(f − , ), whose radius converge to zero. Each sequence will have associated a suitable sequence of Möbius transformations as follows.
Part I. Building sequences of half-circles at the left of C(f , ) and at right of C(f − , ). We divide the closed intervalsÎ , and −Î , ⁴ into six as is shown in Figure 11, and we consider the second sixth closed subintervals of them, which are given by L , = +s + · , +s + · = , and −L , = − , − , respectively. Now, we write L , and −L , as the union of closed subintervals following ,m  (α , ) − ,m , (α , ) ,m and (α , ) − ,m respectively, and the same radius (r( ))m = · · m (see Figure 12). Now, we take the Möbius transformations On the other hand, for every m ∈ N, we consider the Möbius transformations having as isometric circles C(f , ) ,m and C(f , ) − ,m , respectively. We note that (a , ) . Then, we de ne the ,m Then we let C(f , ) ,m , C(f , ) − ,m , C(f , ) ,m , C(f , ) − ,m be the half-circles having centers (α , ) ,m , (α , ) − ,m , (α , ) ,m and (α , ) − ,m respectively, and the same radius (r( ))m = · · m (see Figure 13). Now, for every m ∈ N we take the Möbius transformations In a similar way, for every m ∈ N we take the Möbius transformations having as isometric circles C(f , ) ,m and C(f The elements of J are hyperbolic and the half-circles of C are pairwise disjoint. For n. Building the set Jn containing in nitely countable Möbius transformation and the set Cn composed by its respective isometric circles. We let C(f n,k ), C(f − n,k ) be the half-circles having centers α n,k and α − n,k respectively (see equation (8) which has as isometric circles C(f n,k ) and C(f − n,k ) are a n,k = − · ( n · + ( + s k− ) + ), c n,k = · n · , d n,k = − · ( n · + ( + s k− ) + ), and b n,k = ·( n · + ( +s k− )+ ) − · n · (see Figure 14). Now, for each k ∈ { , . . . , n− − } we shall build sequences of half-circles at the left and at the right of C(f n,k ), C(f − n,k ), whose radius converge to zero, each sequence will have associated a suitable sequence of Möbius transformations as follows.