

CUBO, A Mathematical Journal
Vol.22, No¯ 01, (39–53). April 2020

http: // doi. org/ 10. 4067/ S0719-06462020000100039

A sufficiently complicated noded Schottky group of rank
three

Rubén A. Hidalgo
1

Departamento de Matemática y Estad́ıstica,

Universidad de La Frontera, Temuco, Chile.

ruben.hidalgo@ufrontera.cl

ABSTRACT

In 1974, Marden proved the existence of non-classical Schottky groups by a theoretical

and non-constructive argument. Explicit examples are only known in rank two; the

first one by Yamamoto in 1991 and later by Williams in 2009. In 2006, Maskit and

the author provided a theoretical method to construct non-classical Schottky groups in

any rank. The method assumes the knowledge of certain algebraic limits of Schottky

groups, called sufficiently complicated noded Schottky groups. The aim of this paper

is to provide explicitly a sufficiently complicated noded Schottky group of rank three

and explain how to use it to construct explicit non-classical Schottky groups.

RESUMEN

En 1974, Marden demostró la existencia de grupos de Schottky no-clásicos con un ar-

gumento teórico y no-constructivo. Se conocen ejemplos expĺıcitos solo en rango dos;

el primero por Yamamoto en 1991 y después por Williams en 2009. En 2006, Maskit

y el autor entregaron un método teórico para construir grupos de Schottky no-clásicos

en cualquier rango. El método asume el conocimiento de ciertos ĺımites algebraicos de

grupos de Schottky, llamados grupos de Schottky anodados suficientemente complica-

dos. El objetivo de este paper es dar un grupo de Schottky anodado suficientemente

complicado expĺıcitamente de rango tres y explicar cómo usarlo para construir grupos

de Schottky no-clásicos expĺıcitos.

Keywords and Phrases: Riemann surfaces; Schottky groups.

2010 AMS Mathematics Subject Classification: 30F40, 30F10.
1Partially supported by Projects Fondecyt 1190001.

http://doi.org/10.4067/S0719-06462020000100039


40 Rubén A. Hidalgo CUBO
22, 1 (2020)

1 Introduction

A Kleinian group G is called a Schottky group of rank g ≥ 2 if it is generated by loxodromic

transformations A1, . . . ,Ag ∈ PSL2(C) such that there is a collection of 2g pairwise disjoint simple

loops α1,α
′
1,α2,α

′
2, . . . ,αg,α

′
g on the Riemann sphere Ĉ, all of them bounding a common domain

D of connectivity 2g, with Aj(αj) = α
′
j and Aj(D) ∩ D = ∅. The above set of generators is called

geometrical and the above loops a fundamental set of loops for G. It is well known that G is a free

group of rank g and that D is a fundamental domain for it. In [3] Chukrow proved that every set

of g generators of G is geometrical. We say that G is a classical Schottky group if it has a set of

geometrical generators, called a classical set of generators, for which we may find a fundamental

set of loops being circles. Classical Schottky groups were firstly considered by Schottky around

1882. In general, a classical Schottky group may have non-classical set of generators.

Examples of classical Schottky groups are given by the finitely generated purely hyperbolic

Fuchsian groups representing a closed surface with holes [2]. Moreover, if such a Fuchsian group

is a two generator group representing a torus with one hole, then every pair of generators is a

classical set of generators [21].

In 1974, Marden [14] provided the existence of non-classical Schottky groups (his proof is

non-constructive). In 1975, Zarrow [27] claimed to have constructed an explicit example of a non-

classical Schottky group of rank two, but it was lately noted by Sato in [22] to be incorrect. The

first explicit (correct) construction was provided by Yamamoto [26] in 1991 and in 2009 another

example was provided by Williams in his Ph.D. Thesis [24], both for rank two. It seems that, for

rank at least three, there is not explicit example in the literature.

In [7], Maskit and the author described a theoretical method to construct non-classical Schot-

tky groups in any rank g ≥ 2. The idea is to consider certain Kleinian groups, obtained as

geometrically finite algebraic limits of Schottky groups of rank g, called sufficiently complicated

noded Schottky groups of rank g (see Section 2). In this paper (see Section 3) we provide an explicit

construction of a sufficiently complicated noded Schottky group of rank three and we used it to

describe how to obtain a infinite family (one-dimensional) non-classical Schottky group of rank

three.

To finish this introduction, and as a matter of completeness, let us mention another conjecture

related to classical Schottky groups. If Ω is the region of discontinuity of a Schottky group G of

rank g, then it is a connected set and Ω/G is a closed Riemann surface of genus g. Conversely, if S

is a closed Riemann surface, then there is a Schottky group G such that S and Ω/G are isomorphic

(Koebe’s uniformization theorem). As we have the existence of non-classical Schottky group, it

might be that G is non-classical. A conjecture (due to Bers) asserts that we may chose G to be

classical. Some positive answers were obtained by Bobenko [1], Koebe [11], Maskit [17], Seppälä

[23] (for the case in which the surface admits antiholomorphic involutions with fixed points) and


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A sufficiently complicated noded Schottky . . . 41

McMullen [13] (for the case in which the surface has sufficiently many shorts geodesics). Recently,

Hou [8, 9] have announced a proof of this conjecture (by using Haussdorf dimension of the limit

set of Kleinian groups) and another approach in [6] (by using Belyi curves).

2 Sufficiently complicated Noded Schottky groups

2.1 Noded Schottky groups

A noded Schottky group of rank g ≥ 2 is geometrically defined as follows. Consider a collection of

pairwise disjoint open topological discs D1, D
′
1, . . . ,Dg and D

′
g on the Riemann sphere Ĉ so that

the corresponding boundaries α̂1 = ∂D1, α̂
′
1 = ∂D

′
1, . . . , α̂g = ∂Dg, α̂

′
g = ∂D

′
g are simple loops

and they only intersect in at most finitely many points. Let Â1, . . . , Âg be Möbius transformations

such that Âj(α̂j) = α̂
′
j and Âj(Dj)∩D

′
j = ∅, for each j = 1, . . . ,g. Observe that the transformation

Âj may only be loxodromic or parabolic. The group Ĝ, generated by these transformations, is

a Kleinian group isomorphic to a free group of rank g. If p is a point of intersection of two of

the above loops, then either it is a fixed point of a parabolic transformation of Ĝ or it has trivial

Ĝ-stabilizer. In the second situation, one may deform in a suitable manner these loops in order

to avoid the intersection at p and not adding extra intersections. In this way, two possibilities

appear (up to performing the above deformation), either: (i) G is a Schottky group of rank g or

(ii) there are intersection points, each of them being a fixed point of a parabolic transformation of

Ĝ. In case (ii) we say that Ĝ is a noded Schottky group of rank g; we call the above set of loops a

fundamental set of loops and the generators a set of geometrical generators.

Remark 1. In [18], as an application of the Klein-Maskit’s combination theorems, it was noted that

a noded Schottky group Ĝ is geometrically finite, that each of its parabolic elements is a conjugate

of a power of one of the transformations fixing a common point of two of the fundamental loops,

and that the complement D̂ of the union of the closures of D1,D
′
1, . . . ,Dg, D

′
g is a fundamental

domain for Ĝ. Different as for the case of Schottky groups, not every set of free generators of a

noded Schottky group is necessarily geometrical.

2.2 The extended region of discontinuity

If Ω is the region of discontinuity of a noded Schottky group Ĝ of rank g, then by adding to it the

parabolic fixed points of Ĝ, and with the appropriate cusped topology (see [5, 12]), we obtain its

extended region of discontinuity Ω+; it happens that S+ = Ω+/Ĝ is a stable Riemann surface of

genus g. In the case that the number of nodes of S+ is 3g − 3, we say that Ĝ is a maximal noded

Schottky group (in this case, there are exactly 2g − 2 connected components of the complement

of the nodes of S+, each one being a triple-punctured sphere). In [4] it was observed that every

stable Riemann surface of genus g is obtained as above; so every point of the Deligne-Mumford


42 Rubén A. Hidalgo CUBO
22, 1 (2020)

Figure 1—A stable Riemann surface of genus 3

with 3 nodes

Figure 2—A stable Riemann surface of genus 3

with 6 nodes

compactification of the moduli space of genus g can be realized by a suitable noded Schottky group

of rank g.

2.3 Neoclassical noded Schottky groups

A noded Schottky group for which there is a set of geometrical generators admitting a fundamen-

tal set of loops all of which are Euclidean circles is called neoclassical; the corresponding set of

generators is called a neoclassical set of generators.

In [7] it was proved that if G is a noded Schottky group such that Ω+/G is a stable Riemann

surface as in Figure 1, then it cannot be neoclassical (this should be still true for every noded

Schottky group of rank g ≥ 4 whose corresponding stable Riemann surface has g +1 components,

one being of genus zero and the others being of genus one).

2.4 Sufficiently complicated noded Schottky groups

The space of deformations of a Schottky group of rank g, denoted by Salg, is a subset of the

representation space of the free group of rank g in PSL2(C), modulo conjugation. Regard H
3 as

being the set {(z,t) : z ∈ C,t > 0 ∈ R}. We likewise identify C with the boundary of H3, except

for the point at infinity; that is, we identify C with {(z,t) : t = 0}.

2.4.1 The relative conical neighbourhood of a noded Schottky group

Let Ĝ be a noded Schottky group of rank g ≥ 2, with a set of geometrical generators Â1, . . . , Âg,

and corresponding fundamental set of loops α̂1, . . . , α̂
′
g, these being the corresponding boundary

loops of a collection of pairwise disjoint open discs D1, . . . ,D
′
g. The complement of the closures of

these discs is a fundamental domain D̂ for Ĝ. Let P̂1, . . . , P̂q be a maximal set of primitive parabolic

elements of Ĝ generating non-conjugate cyclic subgroups, where q ≥ 1 (we may assume the fix point

of these parabolic transformations to be contained in the intersection of two fundamental loops).

We denote by Ω(Ĝ) its region of discontinuity and by Ω+(Ĝ) its extended region of discontinuity.


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A sufficiently complicated noded Schottky . . . 43

Next, we proceed to recall a construction done in [7] of a one-real family of Schottky groups Gτ

whose geometric limit is Ĝ.

(I) The infinite shoebox construction

For each i = 1, . . . ,q, choose a particular Möbius transformation Hi conjugating P̂i to the trans-

formation P(z) = z + 1 and consider the renormalized group HiĜH
−1
i . For this group, there is a

number τ0 > 1 so that the set {| Im(z)| ≥ τ0} is precisely invariant under the stabilizer Stab(∞)

of ∞ in the group HiĜH
−1
i . In this normalization, for each parameter τ, with τ > τ0, we define

the infinite shoebox to be the set B0,τ = {(z,t) : | Im(z)| ≤ τ,t ≤ τ}. Since τ0 > 1, we easily

observe that for every τ > τ0, the complement of B0,τ in H
3 ∪ C is precisely invariant under

Stab(∞) ⊂ HiĜH
−1
i , where we are now regarding Möbius transformations as hyperbolic isome-

tries, which act on the closure of H3. Then for Ĝ, the infinite shoebox with parameter τ at zi, the

fixed point of P̂i, is Bi,τ = H
−1
i (B0,τ). If P̂ is any parabolic element of Ĝ, conjugate to some power

of P̂i, then the corresponding infinite shoebox at the fixed point of P̂, is given by T(Bi,τ), where

P̂ = TP̂iT
−1. It was observed in [19] that, for each fixed τ > τ0, Ĝ acts as a group of conformal

homeomorphisms on the expanded regular set Bτ =
⋂
Â(Bi,τ), where the intersection is taken over

all Â ∈ Ĝ and all i = 1, . . . ,q. Further, Ĝ acts as a (topological) Schottky group (in the sense

of our geometrical definition) on the boundary of Bτ. Each parabolic P̂ ∈ Ĝ appears to have two

fixed points on the boundary of Bτ; that is, P̂, as it acts on the boundary of Bτ, appears to be

loxodromic. The flat part of Bτ is the intersection of Bτ with the extended complex plane Ĉ. The

complement of the flat part (on the boundary of Bτ) is the disjoint union of 3-sided boxes, where

each box has two vertical sides (translates of the sets {Im(z) = ±τ,0 < t < τ}) and one horoball

side (a translate of the set {| Im(z)| ≤ τ,t = τ}). For each i = 1. . . ,q and for each integer n ≥ 1,

we set Bi,τ,n = H
−1
i (B0,τ ∩ {| Re(z)| ≤ n}) and B

τ,n =
⋂

Â∈Ĝ Â(Bi,τ,n). The truncated flat part

of Bτ,n is the intersection Bτ,n ∩ Ĉ. The boundary of the truncated flat part near a parabolic

fixed point, renormalized so as to lie at ∞, is a Euclidean rectangle. Let us renormalize Ĝ so that

∞ ∈ Ω(Ĝ). Then, for each τ > τ0, there is a conformal map f
τ, mapping the boundary of Bτ

to Ĉ, and conjugating Ĝ onto a Schottky group Gτ, where fτ is defined by the requirement that,

near ∞, fτ(z) = z+O(|z|−1). The group Gτ depends on the choice of the Möbius transformations

H1, . . . ,Hq as well as on the choice of τ. The elements A
τ
1 = f

τÂ1(f
τ)−1, . . . ,Aτg = f

τÂg(f
τ)−1

provide a set of free generators for the Schottky group Gτ.

(II) Vertical projection loops

Next, we proceed to construct a fundamental set of loops for Gτ for the above generators in terms

of the fundamental set of loops for Ĝ. In [19] it was shown that, with the above normalization,

fτ converges to the identity I, uniformly on compact subsets of Ω(Ĝ), and, for each j ∈ {1, . . . ,g},

Aτj converges to Âj, as τ → ∞. In particular, if we fix τ0, and fix n, then f
τ → I uniformly on


44 Rubén A. Hidalgo CUBO
22, 1 (2020)

compact subsets of the truncated flat part of Bτ0,n. The boundary of Bτ0,n consists of a disjoint

union of quadrilaterals with circular sides. After renormalization, the part of the boundary of

Bτ0,n corresponding to {| Im(z)| = τ0} is the horizontal part of the boundary, while the part of

the boundary corresponding to {| Re(z)| = n} is the vertical part of the boundary. We make a

fixed choice of the conjugating maps, Hi, i = 1, . . . ,q, and we fix a choice of the parameter

τ > τ0 in the above construction. We may deform all the loops α̂i and α̂
′
i, within Ω

+(Ĝ) to an

equivalent fundamental set of loops, with the same geometric generators Â1, . . . , Âg, so that, after

appropriate renormalization, each connected component of each of the deformed loops appears,

in each component of the complement of the flat part of Bτ, as a pair of half-infinite Euclidean

vertical lines, one in {Im(z) ≥ τ}, the other in {Im(z) ≤ −τ}, both with the same real part (the

technical details of such a deformation can be found in [7]). We call such a deformed loops the

vertical projection loops. These vertical projection loops, which we still denoting as α̂1, . . . , α̂
′
g,

yields a fundamental set of loops, ατ1, . . . ,α
′
g
τ
for the generators Aτ1, . . . ,A

τ
g of the Schottky group

Gτ.

(III) The relative conical neighborhood

The relative conical neighborhood of Ĝ is to be defined as the set of all marked Schottky groups

Gτ = 〈Aτ1, . . . ,A
τ
g〉, with the fundamental set of loops α

τ
1, . . . ,α

′
g
τ
, as constructed above.

Remark 2. Recall that we are assuming that ∞ is an interior point of the flat part corresponding

to τ0, and f
τ(z) = z+O(|z|−1) near ∞. As, with these normalizations, fτ → I uniformly on compact

subsets of Ω(Ĝ), we obtain that Gτ → Ĝ algebraically. It now follows, from the Jørgensen-Marden

criterion [10], that Gτ → Ĝ geometrically and that each relative conical neighborhood contains

infinitely many distinct marked Schottky groups. It is also easy to see, as in [19], that, for each

primitive parabolic element P̂ ∈ Ĝ, as τ → ∞, there is a corresponding geodesic on Sτ = Ω(Gτ)/Gτ

whose length tends to zero. It follows that each relative conical neighborhood of a noded Schottky

group contains Schottky groups representing infinitely many distinct Riemann surfaces.

2.4.2 Pinchable loops of Schottky groups

Let G be a Schottky group of rank g ≥ 2, with generators A1, . . . ,Ag, and let π : Ω(G) → S be a

regular covering with deck group G.

(IV) Pinchable loops

Let γ1, . . . ,γq be a set of simple disjoint geodesics loops on S. Each γj corresponds to a conjugacy

class of a cyclic subgroup of G (including the trivial subgroup) by the lifting under π; let 〈Wj〉

be a representative of such a class. If these q cyclic subgroups are non-trivial, they are pairwise

non-conjugated in G and the generators Wj are non-trivial powers in G (i.e., there is no Tj ∈ G so


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A sufficiently complicated noded Schottky . . . 45

that Wj = T
mj
j for some mj ≥ 2), then we say that this set of geodesics is pinchable in G.

Remark 3. (1) It was shown in [20] (see also Yamamoto [25]) that if γ1, . . . ,γq is a set of pinchable

simple disjoint geodesics loops on S in G, defined by the words W1, . . . ,Wq, as above, then there is

a noded Schottky group Ĝ, and there is an isomorphism ψ : G → Ĝ, where ψ(W1), . . . ,ψ(Wq), and

their powers and conjugates, are exactly the parabolic elements of Ĝ. More precisely, it was shown

in [20] that there is a path in Schottky space, Salg, which converges to a set of generators for Ĝ,

along which the lengths of the geodesics, γ1, . . . ,γq, all tend to zero. (2) On the other direction,

let us consider a noded Schottky group Ĝ of rank g ≥ 2, with a set of geometrical generators

Â1, . . . , Âg and corresponding fundamental set of loops α̂1, . . . , α̂
′
g. Let G

τ be a Schottky group

of rank g with fundamental set of loops ατ1, . . . ,α
′
g
τ
and generators Aτ1, . . . ,A

τ
g, in a relative

conical neighborhoodof Ĝ as previously described in Section 2.4.1. Let S = Ω(Gτ)/Gτ be the

closed Riemann surface of genus g represented by Gτ, and let Vi ⊂ S be the projection of α
τ
i ,

i = 1, . . . ,g. Then V1, . . . ,Vg is a set of g homologically independent simple disjoint loops on S.

Let ψ : Gτ → Ĝ be the isomorphism defined by Aτi 7→ Âi, i = 1, . . . ,g. The elements of G
τ which

are sent to parabolic elements of Ĝ are called the pinched elements of Gτ. There are simple disjoint

geodesics γ1, . . . ,γq on S, defined by pinched elements of G
τ, given by the words W1, . . . ,Wq in

the generators Aτ1, . . . ,A
τ
g, so that every parabolic element of Ĝ is a power of a conjugate of one

of their ψ-image. It happens that this collection of loops γ1, . . . ,γq is a set of pinchable geodesics

of Gτ. The construction in [19] shows that we can choose the above parameter τ so that the γi

are all arbitrarily short.

Proposition 1 ([7]). Every non-empty set of k < 3g − 3 pinchable geodesics is contained in a

maximal set of 3g − 3 pinchable geodesics.

(V) Valid sets of fundamental loops and their complexity

Let γ1, . . . ,γq ⊂ S be a pinchable set of geodesics in G. Set Ŝ
+ the stable Riemann surface

obtained from S by pinching these q geodesics; it consists of a finite number of compact Riemann

surfaces, called parts, which are joined together at a finite number of nodes. Also, let Ĝ be the

noded Schottky group obtained from G by pinching these q geodesics.

Let V1, . . . ,Vg, be a fundamental set of loops for G on S (that is, the components of the lifting

of these loops under π are simple loops and such a lifting set of loops contains a fundamental set of

loops for G) and let V̂1, . . . , V̂g be the corresponding loops on Ŝ
+ obtained by pinching γ1, . . . ,γq.

We observe that the lifts of the V̂i to Ω
+(Ĝ) are all loops, but they are generally not disjoint and

they need not to be simple. There are certainly some number of these lifts passing through each

parabolic fixed point, and some of them might pass more than once through the same parabolic

fixed point. The set of loops, V1, . . . ,Vg, is called a valid set of fundamental loops for γ1, . . . ,γq,

if every lift of every V̂i to Ω
+(Ĝ) is a simple loop; that is, it passes at most once through each

parabolic fixed point (i.e., the set of loops, V̂1, . . . , V̂g, forms a fundamental set of loops for Ĝ on


46 Rubén A. Hidalgo CUBO
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Ŝ+). We note that there are exactly q equivalence classes of parabolic fixed points in Ĝ, one for

each of the loops γi.

Proposition 2 ([4, 7]). There is at least one valid set of fundamental loops V1, . . . ,Vg, for every

set of pinchable geodesics, γ1, . . . ,γq.

(VI) The complexity

Let us now consider a valid set of fundamental loops, V1, . . . ,Vg, for a set of pinchable geodesics

γ1, . . . ,γq. We can deform the Vi on S so that they are all geodesics. Then the geometric

intersection number, Vi •γj, of Vi with γj is well defined; it is the number of points of intersection

of these two geodesics. Looking on the corresponding noded surface Ŝ+, Vi • γj is the number

of times the curve V̂i obtained from Vi by contracting γj to a point, passes through that point

(node). The complexity of V1, . . . ,Vg, with respect to γ1, . . . ,γq, is given by

Ξ(γ1, . . . ,γq;V1, . . . ,Vg) = max
1≤j≤q

g∑

i=1

Vi • γj,

and the complexity Ξ(γ1, . . . ,γq) is the minimum of Ξ(γ1, . . . ,γq;V1, . . . ,Vg), where the minimum

is taken over all valid sets of fundamental loops. If Ξ(γ1, . . . ,γq) ≥ n, then, for every valid

fundamental set V1, . . . ,Vg, there is a node P on S
+ so that the total number of crossings of P by

V̂1, . . . , V̂g is at least n.

Proposition 3 ([7]). Let g ≥ 2 and G be a Schottky group of rank g. For each positive integers

n there are only finitely many topologically distinct maximal (i.e. q = 3g − 3) pinchable set of

geodesics in G and complexity n. In particular, there are infinitely many topologically distinct

maximal noded Schottky groups of rank g and there are only finitely many topologically distinct

maximal neoclassical noded Schottky groups in each rank g.

(VII) Sufficiently complicated pinchable sets of geodesics

Now, we consider a maximal set of pinchable geodesics in G, say γ1, . . . ,γ3g−3; so Ĝ is a maximal

noded Schottky group. Observe that Ĝ is rigid, and that every part of S+ is a sphere with

three distinct nodes. Also, every connected component ∆ ⊂ Ω(Ĝ) is a Euclidean disc ∆, where

∆/Stab(∆) is a sphere with three punctures; the three punctures correspond to the three nodes of

the corresponding part of Ŝ+.

Let V1, . . . ,Vg be a valid set of fundamental loops on S for the given set of pinchable geodesics

(the existence is given by Proposition 2), and let V̂1, . . . , V̂g be the corresponding loops on Ŝ
+.

For each i = 1, . . . ,g, the intersection of a lifting of V̂i with a component of Ĝ (i.e., a connected

component of Ω(Ĝ)) is called a strand of that lifting V̂i. Similarly, the loops V̂1, . . . , V̂g appear

on the corresponding parts of Ŝ+ as collections of strands connecting the nodes on the boundary


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A sufficiently complicated noded Schottky . . . 47

of each part. There are two possibilities for these strands; either a strand connects two distinct

nodes on some part, or it starts and ends at the same node. Since the loops V1, . . . ,Vg are simple

and disjoint, there are at most three sets of parallel strands of the V̂i in each part; that is, there

are at most three sets of strands, where any two strands in the same set are homotopic arcs with

fixed endpoints at the nodes. We regard each of these sets of strands on a single part as being

a superstrand, so that there are at most 3 superstrands on any one part. We next look in some

component ∆ of Ω(Ĝ), and look at a parabolic fixed point x on its boundary, where x corresponds

to the node N on the part Si of Ŝ
+. In general, there will be infinitely many liftings of superstrands

emanating from x in ∆, but, modulo Stab(∆) there are only finitely many. In fact, there are at most

4 such liftings of superstrands emanating from x. If there is exactly one superstrand on Si with

one endpoint at N, and the other endpoint at a different node, then modulo Stab(∆) there will be

exactly the one lifting of this superstrand emanating from x. If there is only one superstrand on Si

with both endpoints at the same node N, then this superstrand has two liftings starting at x, one in

each direction; so, in this case, we see two lifts of superstrands modulo Stab(∆) emanating from x.

It follows that, modulo Stab(∆), we can have 0, 1, 2, 3 or 4 liftings of superstrands starting at x. We

note that these liftings of superstrands all end at distinct parabolic fixed points on the boundary

of ∆. We say that the fundamental set of loops, V̂1, . . . , V̂p is sufficiently complicated if there are

two (different) lifts α̂i and α̂j, of some V̂i and some not necessarily distinct V̂j, respectively, so

that α̂i and α̂j both pass through the parabolic fixed point z1, into a component ∆1 of Ĝ, then

both travel through ∆1 to the same parabolic fixed point on its boundary, z2, and into another

component ∆2, which they again traverse together to the same boundary point, z3, necessarily a

parabolic fixed point, where they enter ∆3, and they leave ∆3 at different parabolic fixed points.

2.4.3 Sufficiently complicated noded Schottky groups

A maximal noded Schottky group Ĝ is sufficiently complicated if every set of valid fundamental

loops on Ŝ+ is sufficiently complicated. We note that (keeping the notation of last section), inside

∆1, α̂i and α̂j are disjoint; they both enter ∆1 at the same point, and they both leave ∆1 at the same

point; hence they cannot both be circles. In [7] the following result, stating a sufficient condition

in terms of the complexity for a maximal noded Schottky group to be sufficiently complicated, was

obtained.

Theorem 1 ([7]). If a maximal noded Schottky group Ĝ has complexity at least 11, then it is

sufficiently complicated.

The previous theorem, together Proposition 3, asserts the existence of infinitely many topo-

logically different sufficiently complicated maximal noded Schottky groups in every rank g ≥ 2.

The following result states sufficient conditions for a Schottky group to be non-classical.

Theorem 2 ([7]). Let Ĝ be a maximal noded Schottky group.


48 Rubén A. Hidalgo CUBO
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(1) If Ĝ is sufficiently complicated, then, for τ sufficiently large, the Schottky group Gτ in the

relative conical neighborhood of Ĝ is non-classical.

(2) If S+ = Ω+(Ĝ)/Ĝ is the stable Riemann surface as shown in figure 2, then Ĝ is sufficiently

complicated.

3 Explicit construction of a sufficiently complicated noded

Schottky group

In this section we construct explicitly a noded Schottky group as in part (2) of Theorem 2, so

part (1) of the same theorem asserts that any Schottky group sufficiently near to Ĝ is necessarily

non-classical.

3.1 A family Schottky groups of rank three

Let L0 be the unit circle, L1 be the real line, L2 be the line through the points 0 and w0 = e
πi/3,

and set (see Figure 7)

F =
{

(p,r) : 1/2 < p < 1, 0 < r < r
∗

(p) :=

√

1 + p2 + p4 + p2 − 1
√
3p

}

.

For each (p,r) ∈ F we set L3 to be the circle with center at 0 and radius r and L4 to be

the circle orthogonal to L0, intersecting L1 at the points p and 1/p with angle π/3 (see Figure 3).

The circle L4 has its center at c =
(

1+p2

2p

)
+ i√

3

(
1−p2

2p

)
and it has radius R = 2√

3

(
1−p2

2p

)
. The

condition p > 1/2 asserts that L2 and L4 are disjoint (tangency occurs when p = 1/2) and the

condition r < r∗(p) asserts that L3 and L4 are disjoint (tangency occurs when r = r
∗(p)). Let τj

be the reflection on Lj, for j = 0,1,2,3,4, so

τ0(z) = 1/z, τ1(z) = z, τ2(z) = w
2z, τ3(z) = r

2/z, τ4(z) =
cz − 1

z − c
,

and let Kr,p = 〈τ0,τ1,τ2,τ3,τ4〉. It turns out that Kr,p is an extended Kleinian group with

connected region of discontinuity Ωr,p and so that Ωr,p/Kr,p is a closed disc with 5 branch values,

of orders 2, 2, 2, 2 and 3, on its border. As a consequence of the Klein-Maskit combination

theorems [18], the group Kr,p has no parabolic transformations, its limit set is a Cantor set and it

is geometrically finite. If we set W = τ2τ1, J = τ0τ1 and L = τ1τ4, then W
3 = L3 = J2 = (WJ)2 =

(LJ)2 = 1.

Now, if A1 = L
−1W−1 = τ4τ2, A2 = τ1A1τ1, and A3 = τ0τ3, so

A1(z) =
cw0z − 1

w0z − c
, A2(z) =

cw20z − 1

w20z − c
, A3(z) = r

2z,


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A sufficiently complicated noded Schottky . . . 49

τ

1

2

3
4

0

ττ

τ

τ

Figure 3—A set of lines and circles

1A

A

A

LW
3

2

Figure 4—The Schottky group Gr,p of rank 3:

the six darkest loops are a fundamental set of

loops

then we set Gr,p = 〈A1,A2,A3〉. In Figure 4 we show the situation for values of p near to 1 and

r near to 0; in which case Gr,p turns out to be a classical Schottky group of rank three.

Lemma 1. If (p,r) ∈ F, then Gr,p is a Schottky group of rank three.

Proof. It can be seen that Gr,p is a finite index normal subgroup of Kr,p and

Kr,p/Gr,p = 〈τ0 : τ
2
0 = 1〉 × 〈τ1,τ2 : τ

2
1 = τ

2
2 = (τ2τ1)

3
= 1〉 ∼= Z2 ⊕ D3.

In particular, Gr,p has the same region of discontinuity as for Kr,p (so a function group),

it is geometrically finite and does not have parabolic transformations. As any of the elliptic

transformations of Kr,p goes into an element of the same order in the quotient Kr,p/Gr,p, we

also have that Gr,p is torsion free. Now, as a consequence of the classification of function groups

[15, 16], the group Gr,p is a Schottky group of rank three (in Figure 4 there is shown a fundamental

set of loops).

The closed Riemann surface Sr,p = Ωr,p/Gr,p of genus 3 admits the group Z2 ⊕ D3 as a

group of conformal/anticonformal automorphisms. On Sr,p we have simple closed curves γ1,..., γ6

which are pinchable (see Figures 5 and 6) with respect to the Schottky group Gr,p; these pinchable

curves correspond to the conjugacy classes of cyclic groups of Gr,p as follows:

γ1 corresponds to A
−1
2 ; γ2 corresponds to A1; γ3 corresponds to A

−1
1 A2;

γ4 corresponds to A
−1
1 A2A

−1
3 A

−1
2 A1A3; γ5 corresponds to A

−1
2 A

−1
3 A2A3;

γ6 corresponds to A1A
−1
3 A

−1
1 A3.


50 Rubén A. Hidalgo CUBO
22, 1 (2020)

3

γ

γ

γ

γ

γ

γ

γ

γ

γ

γ

1

2

3
γ

4

4

4 γ

4

5

5
5

5

6
6

6
γ

γ
γ

6

γ

Figure 5—A set of pinchable curves seen at

the Schottky uniformization

5

α

α
α

γ

γ

γ

γ

γ

2

1

3

3

1
2

6

4

γ

Figure 6—A set of pinchable curves seen on

the Riemann surface Sr,p

3.2 A sufficiently complicated noded Schottky groups of rank three

To obtain the desired noded Schottky group, we need to move the pair (p,r) ∈ F to some

point in the boundary in order to have that the loxodromic transformations A−12 , A1, A
−1
1 A2,

A−11 A2A
−1
3 A

−1
2 A1A3, A

−1
2 A

−1
3 A2A3 and A1A

−1
3 A

−1
1 A3 are transformed into parabolic transfor-

mations. As the order three automorphism W permutes cyclically γ1,γ2,γ3 and also γ4,γ5,γ6,

we only need to take care of A1 and A1A
−1
3 A

−1
1 A3. First, in order to transform A1 into a parabolic

transformation we only need to have τ4(w0) = w0, equivalently, that the circle L4 is tangent to

the line L2 at w0. This happens exactly when p = 1/2. Now, assuming p = 1/2, in order for

A1A
−1
3 A

−1
1 A3 to be a parabolic transformation we only need to have tangency of the circle L3

with L4, that is, r = r
∗(1/2) =

√
7−

√
3

2
. The group Gr∗(1/2),1/2 turns out to be a noded Schottky

group that uniformizes a stable Riemann surface as shown in Figure 2 and, by (2) in Theorem 2,

it is a sufficiently complicated noded Schottky group.

Non−classical Schottky groups

r=r *(p)

p

r

p=1/2 p=1p0

(1/2)*r

r
0

Figure 7—The region F and the filled part for the non-classical Schottky groups

3.3 Non-classical Schottky groups of rank three

By (1) in Theorem 2, there exist (p0,r0) ∈ F with the property that if (p,r) ∈ F, 1/2 < p < p0

and r0 < r < r
∗(1/2), then Gr,p is a non-classical Schottky group of rank three (see filled part

region in Figure 7). Moreover, each of these Schottky groups is contained in a Kleinian group Kr,p


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A sufficiently complicated noded Schottky . . . 51

as a finite index normal subgroup with Kr,p/Gr,p ∼= Z2 ⊕ D3, in other words, the closed Riemann

surfaces Sr,p = Ω(Gr,p)/Gr,p admit a group of conformal automorphisms isomorphic to Z2 ⊕D3.

The family of these Riemann surfaces degenerates to a stable Riemann surface Sr∗(1/2),1/2 as

Figure 2 keeping the above group of automorphisms invariant.


52 Rubén A. Hidalgo CUBO
22, 1 (2020)

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	Introduction
	Sufficiently complicated Noded Schottky groups
	Noded Schottky groups
	The extended region of discontinuity
	Neoclassical noded Schottky groups
	Sufficiently complicated noded Schottky groups
	The relative conical neighbourhood of a noded Schottky group
	Pinchable loops of Schottky groups
	Sufficiently complicated noded Schottky groups


	Explicit construction of a sufficiently complicated noded Schottky group
	A family Schottky groups of rank three
	A sufficiently complicated noded Schottky groups of rank three
	Non-classical Schottky groups of rank three