

CUBO A Mathematical Journal
Vol.19, No¯ 03, (69–77). October 2017

Totally Degenerate Extended Kleinian Groups
Rubén A. Hidalgo

1

Departamento de Matemática y Estad́ıstica

Universidad de La Frontera, Temuco, Chile

ruben.hidalgo@ufrontera.cl

ABSTRACT

The theoretical existence of totally degenerate Kleinian groups is originally due to

Bers and Maskit. In fact, Maskit proved that for any co-compact non-triangle Fuch-

sian group acting on the hyperbolic plane H2 there is a totally degenerate Kleinian

group algebraically isomorphic to it. In this paper, by making a subtle modification

to Maskit’s construction, we show that for any non-Euclidean crystallographic group

F, such that H2/F is not homeomorphic to a pant, there exists an extended Kleinian

group G which is algebraically isomorphic to F and whose orientation-preserving half

is a totally degenerate Kleinian group. Moreover, such an isomorphism is provided

by conjugation by an orientation-preserving homeomorphism φ : H2 → Ω, where Ω

is the region of discontinuity of G. In particular, this also provides another proof to

Miyachi’s existence of totally degenerate finitely generated Kleinian groups whose limit

set contains arcs of Euclidean circles.

RESUMEN

La existencia teórica de grupos Kleinianos totalmente degenerados se debe original-

mente a Bers y Maskit. De hecho, Maskit demostró que para cualquier grupo Fuchsiano

co-compacto y no-triangular actuando en el plano hiperbólico H2 existe un grupo Kleini-

ano totalmente degenerado algebraicamente isomorfo a él. En este art́ıculo, haciendo

una modificación sutil a la construcción de Maskit, mostramos que para cualquier grupo

cristalográfico no-Euclidiano F tal que H2/F no es homeomorfo a un pantalón, existe

un grupo Kleiniano extendido G que es algebraicamente isomorfo a F y cuya mitad

que preserva orientación es un grupo Kleiniano totalmente degenerado. Más aún, un

tal isomorfismo está dado por la conjugación por un homeomorfismo que preserva ori-

entación φ: H2 → Ω, donde Ω es la región de discontinuidad de G. En particular, esto

1 Partially supported by Project Fondecyt 1150003 and Anillo ACT 1415 PIA-CONICYT


70 Rubén A. Hidalgo CUBO
19, 3 (2017)

también entrega otra demostración del resultado de Miyachi acerca de la existencia de

grupos Kleinianos totalmente degenerados finitamente generados cuyo conjunto ĺımite

contiene arcos de circunferencias Euclidianas.

Keywords and Phrases: Kleinian Groups, NEC groups

2010 AMS Mathematics Subject Classification: 30F40, 30F50


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Totally Degenerate Extended Kleinian Groups. 71

1 Introduction

The classical uniformization theorem asserts that every non-exceptional Riemann surface S, i.e.,

non-isomorphic to either the Riemann sphere Ĉ, the complex plane C, the puncture plane C − {0}

or a torus, is conformally equivalent to a quotient H2/Γ, where Γ ∼= π1(S,∗) is a discrete group

of conformal automorphisms of the hyperbolic plane H2. Similarly, every Klein surface (i.e., a

real surface where the local change of coordinates are either conformal or anti-conformal) which

are non-exceptionals (i.e., those whose orientation-preserving doble covers are non-exceptional

Riemann surfaces) is di-analytically equivalent to a quotient X = H2/F, where F ∼= πorb1 (X) is a

discrete group of conformal and anti-conformal automorphisms of H2 (necessarilly containing anti-

conformal ones). If X is compact and F+ is the index two subgroup of F consisting of its conformal

elements, then F is called a non-euclidean crystallographic (NEC) group of algebraic genus equal

to the genus of the closed Riemann surface X+ = H2/F+. If X is homeomorphic to the closure of

the complement of three disjoint closed discs on the Riemann sphere, then it is called a (compact)

pant.

A finitely generated non-elementary Kleinian group with a non-empty connected and simply-

connected region of discontinuity is called totally degenerate. The theoretical existence of such

groups, in the boundary of Teichmüller spaces of co-compact non-triangle Fuchsian groups, is by

now well-know due to Bers [1] and Maskit [4], but it seems that there is no explicit example of

such type of groups in the literature. There is a nice construction in [3, IX.G] due to Maskit, as an

application of the Klein-Maskit combination theorems [5, 6, 7] to (theoretically) obtain a totally

degenerate Kleinian group isomorphic to a given co-compact Fuchsian group F+, different from a

triangle one (i.e., the uniformized orbifold is not the sphere with exactly there cone points). The

idea was to observe the existence of some ρ0 ∈ (0,+∞) (which can be chosen in the complement

of a suitable countable subset), an open arc T of the circle centered at the origin and radius ρ0,

with ρ0 ∈ T, such that:

(i) for each t ∈ T, there is a quasifuchsian group G+(t) and there is an isomorphism θt : F
+
→

G+(t),

(ii) G+(ρ0) is a Fuchsian group, and

(iii) for one of the end points t# of T, there is a totally degenerate Kleinian group G+(t#),

with region of discontinuity Ω, so that there is an orientation-preserving homeomorphism

φ : H2 → Ω inducing, by conjugation, an isomorphism between F+ and G+(t#).

An extended Kleinian group is a group G of conformal and anticonformal automorphisms of

the Riemann sphere, necessarily containing anticonformal elements, whose index two orientation-

preserving half G+ is a Kleinian group. In the case that G+ is totally degenerate, we say that

G is a totally degenerate extended Kleinian group. The existence of totally degenerate extended

Kleinian groups should not be a surprise as, from a general abstract point of view, this follows


72 Rubén A. Hidalgo CUBO
19, 3 (2017)

from the non-triviality of the Teichmuller space of an orbifold with mirrored boundary. In this

note we indicate the subtle modifications in Maskit’s construction from [3, IX.G], to observe that

for given a NEC group there is a totally degenerate Kleinian group isomorphic to it.

Theorem 1.1. Let F be a NEC group so that H2/F is not homeomorphic to a pant. Then there ex-

ists a totally degenerate extended Kleinian group G, with region of discontinuity Ω, and there exists

an orientation-preserving homeomorphism φ : H2 → Ω inducing, by conjugation, an isomorphic

between F and G.

If for the NEC group F it holds that H2/F has non-empty boundary, then Theorem 1.1 implies

the following result due to Miyachi.

Corollary 1.2 (Miyachi [8]). There are totally degenerate Kleinian groups for which there exist

arcs of Euclidean circles inside the limit set; these arcs connect fixed points of distinct parabolic

transformations and/or connect the fixed points of the same hyperbolic transformation. These arcs

of circles are dense in the limit set.

2 Preliminaries

2.1 Möbius and extended Möbius transformations

The conformal automorphisms of the Riemann sphere Ĉ are the Möbius transformations and its

anti-conformal ones are the extended Möbius transformations (the composition of the standard

reflection J(z) = z with a Möbius transformation). We denote by M the group of Möbius transfor-

mations and by M̂ the group generated by M and J. Clearly, M is an index two subgroup of M̂. If

K is a subgroup of M̂, then we set K+ := K ∩ M.

Möbius transformations are classified into parabolic, loxodromic (including hyperbolic) and

elliptic transformations. First, we need to observe that a non-trivial Möbius transformation has at

least one fixed point and at most two of them. The parabolic ones are those having exactly one fixed

point, elliptic ones are conjugated to rotations and loxodromic are conjugated to transformations

of the form z 7→ reiθz, where r ∈ (0,1) ∪ (1,+∞) (if eiθ = 1, then we call it hyperbolic). Similarly,

extended Möbius transformations are classified into pseudo-parabolic (the square is parabolic),

glide-reflection (the square is hyperbolic), pseudo-elliptic (the square is elliptic), reflection (of order

two admitting a circle of fixed points on Ĉ) and imaginary reflection (of order two and having no

fixed points on Ĉ) [3].

Each Möbius transformation γ can be identified with a projective linear transformation

γ =

[
a b

c d

]
∈ PSL2(C)

and the square of its trace tr(γ)2 = (a + d)2 is well defined. If γ is different from the identity

transformation, then the following hold:


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Totally Degenerate Extended Kleinian Groups. 73

(1) γ is parabolic if and only if tr(γ)2 = 4.

(2) γ is elliptic if and only if tr(γ)2 ∈ [0,4).

(3) γ is loxodromic if and only if tr(γ)2 /∈ [0,4] (hyperbolic ones correspond to tr(γ)2 ∈ (4,+∞)).

2.2 Kleinian and extended Kleinian groups

A Kleinian group is a discrete subgroup of M, and an extended Kleinian group is a discrete subgroup

of M̂ containing extended Möbius transformations.

If G is either a Kleinian or an extended Kleinian group, then its region of discontinuity Ω(G) is

the open subset of Ĉ (which might be empty) formed by those points p ∈ Ĉ with finite G-stabilizer

Gp = {γ ∈ G : γ(p) = p} and for which there is an open set Up, p ∈ Up, such that γ(Up)∩Up = ∅,

for γ ∈ G − Gp. The complement Λ(G) = Ĉ − Ω(G) is called the limit set of G. If the limit set

is finite, then G is called elementary; otherwise, it is called non-elementary. If G is an extended

Kleinian group, then both G and G+ have the same region of discontinuity.

Basic examples of Kleinian groups are the following ones. A function group is a finitely gener-

ated Kleinian group G+ with an invariant connected component ∆ of its region of discontinuity.; in

this case, Λ(G+) = ∂∆. A quasifuchsian group is a function group whose limit set is a Jordan loop.

A B-group is a finitely generated function group with a simply connected invariant component of

its region of discontinuity. If G+ is a B-group, say with the simply-connected invariant connected

component ∆, then the Riemann mapping’s theorem ensures the existence of a biholomorphism

F : H2 → ∆. Then F−1G+F is a discrete group of automorphisms of H2 (a Fuchsian group). A

parabolic transformation L ∈ G+ is called accidental if F−1 ◦ L ◦ F is hyperbolic.

Examples of extended Kleinian groups are the following ones. An extended function group is

a finitely generated extended Kleinian group G with an invariant connected component ∆ of its

region of discontinuity. In this case, Λ(G) = ∂∆ and G+ is a function group (the converse of this

last fact is not in general true). An extended quasifuchsian group is an extended function group

whose limit set is a Jordan loop (so its orientation-preserving half is a quasifuchsian group). An

extended B-group is a finitely generated extended Kleinian group with a simply connected invariant

connected component of its region of discontinuity. We observe that the orientation-preserving half

of an extended B-group is a B-group, but the converse is in general not true (see part (2) of the

next result).

Lemma 2.1. If G is an extended Kleinian group whose orientation-preserving half G+ is a B-

group, then either (1) G is an extended B-group, or (2) G+ is a quasifuchsian group and there is

an element of G − G+ permuting both components of its region of discontinuity.

Proof. Since G+ is finitely generated, so is G. Let ∆ be a simply connected invariant component of

G+. If G is not an extended B-group, then there is some γ ∈ G−G+ so that ∆′ = γ(∆) is another


74 Rubén A. Hidalgo CUBO
19, 3 (2017)

different simply connected component of G+. It follows that G+ is necessarily a quasifuchsian

group.

Remark 2.2. Let G be an extended B-group. Then the following properties are easy to see (just

from the previous definitions).

(1) G is an extended quasifuchsian group if and only if G+ is a quasifuchsian group.

(2) G is an extended totally degenerate group if and only if G+ is totally degenerate;

(3) G has accidental parabolic transformations if and only if G+ has accidental parabolic trans-

formations.

The above remark permits us to see the following fact.

Lemma 2.3. Let G be a non-elementary extended B-group. Then either

(1) G is an extended quasifuchsian group, or

(2) G is an extended totally degenerate group, or

(3) G contains accidental parabolic transformations.

Proof. As G+ is a non-elementary B-group, then either G+ is a quasifuchsian group, or G+ is a

totally degenerate group, or G+ contains accidental parabolic transformations. The result now

follows from Remark 2.2.

2.3 The Klein-Maskit’s combination theorem

We next state a simple version of Klein-Maskit’s combination theorems which is enough for us in

this paper.

Theorem 2.4 (Klein-Maskit’s combination theorem [5, 6, 7]).

(1) (Free products) Let Kj be a (extended) Kleinian group with region of discontinuity Ωj, for

j = 1,2. Let Fj be a fundamental domain for Kj and assume that there is a simple closed loop Σ,

contained in the interior of F1 ∩ F2, bounding two discs D1 and D2, so that, for j = 1,2, the set

Σ∪Dj ⊂ Ω3−j is precisely invariant under the identity in K3−j. Then K = 〈K1,K2〉 is a (extended)

Kleinian group, with fundamental domain F1 ∩ F2, which is the free product of K1 and K2. Every

finite order element in K is conjugated in K to a finite order element of either K1 or K2. Moreover,

if both K1 and K2 are geometrically finite, then K is so.

(2) (HNN-extensions) Let K0 be a (extended) Kleinian group with region of discontinuity Ω, and

let F be a fundamental domain for K0. Assume that there are two pairwise disjoint simple closed

loops Σ1 and Σ2, both of them contained in the interior of F0, so that Σj bounds a disc Dj such that


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Totally Degenerate Extended Kleinian Groups. 75

(Σ1 ∪D1)∩(Σ2 ∪D2) = ∅ and that Σj ∪Dj ⊂ Ω is precisely invariant under the identity in K0. If T

is either a loxodromic transformation or a glide-reflection so that T(Σ1) = Σ2 and T(D1)∩D2 = ∅,

then K = 〈K0,f〉 is a (extended) Kleinian group, with fundamental domain F1 ∩ (D1 ∪D2)
c, which

is the HNN-extension of K0 by the cyclic group 〈T〉. Every finite order element of K is conjugated

in K to a finite order element of K0. Moreover, if K0 is geometrically finite, then K is so.

3 Proof of Theorem 1.1

We proceed to describe the main points of the arguments done in [4, IX.G], for the Fuchsian groups

case, and the corresponding adaptation to the NEC groups case.

Let F be a NEC group acting on the upper-half plane H2, such that H2/F is not a pant, and

let π : H2 → H2/F be a di-analytic regular branched covering map induced by the action of F.

As H2/F is not a pant, we may choose a simple loop w ⊂ H2/F so that each of its lifted arcs in

H
2, under π, has as F-stabilizer a cyclic group generated by a hyperbolic element being primitive

(that is, it is not a non-trivial power of an element of F). Let A ⊂ H2 be one of the arcs in

Σ := π−1(w) and let J = 〈j〉 be its F-stabilizer (so j is a primitive hyperbolic transformation). Note

that the connected components of H2 − Σ are planar regions. Let E1 and E2 be the two of these

regions containing A on their borders and, for m ∈ {1,2}, let Fm be the F-stablizer of Em. As Em
is precisely invariant under Fm in F, it follows that Em/Fm is embedded in H

2/F. Moreover, H2/F

is the union of E1/F1, E2/F2 and w. As Em/Fm is topologically finite, Fm is finitely generated. As

a consequence of the first Klein-Maskit combination theorem [5] it holds that

F = 〈F1,F2〉 = F1 ∗J F2.

We may normalize F so that A is contained in the imaginary line, that is, j has its fixed points

at 0 and ∞. We assume that E1 contains positive real points on its border.

If t ∈ C − {0} and kt(z) = tz, then let G(t) = 〈F1,F2(t)〉, where F2(t) = ktF2k
−1
t . For each t

there is a natural surjective homomorphism φt : F → G(t), defined as the identity on F1, and as the

isomorphism f 7→ kt ◦f◦k
−1
t on F2. Clearly, this restricts to a surjective homomorphism φt : F

+
→

G(t)+. We should note that, for t ∈ (1,+∞), the group G(t) is a group of conformal and anti-

conformal automorphisms of H2, in particular, asserting that G(t) is a NEC group topologically

conjugated to F.

In [4, Lemma G.6]) it was noted that, for every f ∈ F+, the fixed points, and the square of

the trace of φt(f), are holomorphic functions of t ∈ C − {0}. As a consequence, for each hyperbolic

f ∈ F+, there are only countable many complex numbers t so that φt(f) ∈ G(t)
+ is parabolic.

Since F+ is countable, there are only countably many ρ > 0 for which there is a t = ρeiθ so that

for some hyperbolic f ∈ F+, φt(f) is parabolic. So, we may find ρ = ρ0 so that tr
2(φt(f)) 6= 4 for

all hyperbolic f ∈ F+ and for all t = ρ0e
iθ.


76 Rubén A. Hidalgo CUBO
19, 3 (2017)

We know from the above that G(ρ0) is still a NEC group and, for θ small, the group G(ρ0e
iθ)

is an extended quasifuchsian group.

Similarly as done for the Fuchsian situation, we let T be the set of complex number of the

form t = ρ0e
iθ for which there is a loop W(t) dividing Ĉ into two closed discs, B1(t) and B2(t),

where B1(t) is a (J,F1)-block (see [4, VII.B.4]) and B2(t) is precisely invariant under J in F2(t).

Note that: (i) ρ0 ∈ T, in this case B1 = B1(ρ0) is the left half-plane and B2 = B2(ρ0) is the right

half-plane, and (ii) for small values of θ, ρ0e
iθ ∈ T.

If t ∈ T, then the groups F1 and F2(t) satisfy the hypothesis to use the first Klein-Maskit

combination theorem, so G(t) = F1 ∗J F2(t) and G(t) is an extended quasifuchsian group so that

φt : F → G(t) is in an isomorphism. Also, there is a homeomorphism ψt : Ω(F) → Ω(G(t))

inducing the isomorphism φt : F → G(t) and so that W(t) is the image under ψt the circle given

as the union of the imaginary line with ∞. Note that B1(t) is the disc containing ψt(E2) and

B2(t) is the disc containing ψt(E1). The limit set Λ(G(t)) is a simple close curve passing through

0 and ∞; the complement, Ω(G(t)) has two components, the upper component ∆(t) and the lower

component ∆′(t).

As a consequence of [3, Proposition IX.G.8], applied to the Fuchsian group F+ and the quasi-

fuchsian group G(t)+, one obtains that, if t ∈ T and Im(t) > 0, then H2 ⊂ ∆(t). Also, as observed

in [4, IX.G.9], by interchanging the roles of F1 and F2(t) it permits to observe that ∆(t) contains

the half-plane {arg(t) < arg(z) < arg(t) + π}. Applying [3, Lemma IX.G.10] to F+ and G(t)+, it

can be seen that T is an open arc of the circle of radius ρ0 with centre at the origin.

Next, we follow [3, IX.G.12]. We start at the point t0 = ρ0 ∈ T and traverse counterclockwise

to reach some first point t# not in T; and we set T0 be the arc of T between t0 and t
# (as noted

in there, as t traverses T0 counterclockwise, from t0 to t
#, the upper component ∆(t) gets larger

and the lower component ∆′(t) gets smaller). Next, we fix a fundamental polygon P1 ⊂ H
2 for

F1 and a fundamental polygon P2 ⊂ H
2 for F2. We choose P1 and P2 so that they are both

contained in some fundamental polygon E for J. Leave P1 fixed and define P2(t) = kt(P2). So

P2(t) is a fundamental polygon for F2(t) in the appropriate half-plane. The union P1 ∪ P2(t)

bounds a fundamental domain D(t) ⊂ ∆(t) for G(t) acting on ∆(t) and there is a homeomorphism

ψt : H
2
→ ∆(t) so that ψt(D(t0)) = D(t) inducing the isomorphism φt : F → G(t).

It follows from the construction that, as t ∈ T0 approaches t
#, φt : F → G(t) converges to a

homomorphism φ : F → G(t#). By our choice on ρ0, it follows from [2] that φ : F
+
→ G(t#)+ is

type-preserving isomorphism.

Lemma 3.1. φ : F → G(t#) is an isomorphism.

Proof. Otherwise, there should be some f ∈ F − F+ so that φ(f) = 1. But, as f2 ∈ F+ and

that φ : F+ → G(t#)+ is an isomorphism, then f2 = 1. As f is an anticonformal involution,

φt(f) ∈ G(t) is an anticonformal involution; so φt(f) cannot approach the identity as t approaches

t#; a contradiction.


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Totally Degenerate Extended Kleinian Groups. 77

Also, as t ∈ T0 approaches t
#, ψt : D(t0) → ψt(D(t0)) ⊂ ∆(t) converges to a homeomorphism

from D(t0) onto its image D. Now, the same proof as [3, Lemma IX.G.13] permits to obtain that

G = G(t#) is an extended Kleinian group and that D is precisely invariant under the identity in

G. Working with F+, G(t)+ and G+, we obtain (from [3, IX.G.14]) that G+ is a B-group with a

simply connected invariant component ∆, where ∆/G+ is a finite Riemann surface homeomorphic

to H2/F+. As consequence of [3, Proposition IX.G.15] it follows that G+ is not quasifuchsian.

Combining [3, IX. G.14], Lemma 2.1 and the above, one obtains that G is an extended B-group,

different from a quasifuchsian one. As ρ0 was constructed so that, for every f ∈ F
+ hyperbolic

and every t = ρ0e
iθ, the element φt(f) ∈ G

+(t) is not parabolic, it follows that φ(f) is neither

parabolic. Hence the only elements of G+(t) that are parabolic are conjugates of the parabolic

elements of F+1 or F2(t)
+. But, as seen in [4, IX.G.14], they represent punctures on ∆/G+(t), so

they are not accidental. Now, as consequence of all the above, together Lemma 2.3, we obtain that

the group G(t#) is an extended totally degenerate group as desired.

References

[1] L. Bers. On boundaries of Teichmüller spaces and on Kleinian groups: I. Ann. of Math. 91

(1970), 570-600.

[2] V. Chuckrow. On Schottky groups with applications to Kleinian groups. Ann. of Math. 88

(1968), 47-61.

[3] B. Maskit, Kleinian Groups, GMW, Springer-Verlag, 1987.

[4] B. Maskit. On boundaries of Teichmüller spaces and on Kleinian groups: II. Ann. of Math.

91 (1970), 607-639.

[5] B. Maskit. On Klein’s Combination Theorem Trans. of the Amer. Math. Soc. 120, No. 3

(1965), 499–509.

[6] B. Maskit. On Klein’s combination theorem III. Advances in the Theory of Riemann Surfaces

(Proc. Conf., Stony Brook, N.Y., 1969), Ann. of Math. Studies 66 (1971), Princeton Univ.

Press, 297-316.

[7] Maskit, B. On Klein’s combination theorem. IV. Trans. Amer. Math. Soc. 336 (1993), 265-294.

[8] H. Miyachi. Quasi-arcs in the limit set of a singly degenerate group with bounded geometry.

In Kleinian Groups and Hyperbolic 3-Manifolds (Eds. Y.Komori, V.Markovic C.Series) LMS.

Lec. Notes 299 (2003), 131-144.


	Introduction
	Preliminaries
	Möbius and extended Möbius transformations
	Kleinian and extended Kleinian groups
	The Klein-Maskit's combination theorem

	Proof of Theorem 1.1