

CUBO, A Mathematical Journal

Vol. 23, no. 03, pp. 369–384, December 2021

DOI: 10.4067/S0719-06462021000300369

The structure of extended function groups

R. A. Hidalgo1

1 Departamento de Matemática y

Estad́ıstica, Universidad de La Frontera,

Temuco, Chile.

ruben.hidalgo@ufrontera.cl

ABSTRACT

Conformal (respectively, anticonformal) automorphisms of the Rie-

mann sphere are provided by the Möbius (respectively, extended

Möbius) transformations. A Kleinian group (respectively, an ex-

tended Kleinian group) is a discrete group of Möbius transforma-

tions (respectively, a discrete group of Möbius and extended Möbius

transformations, necessarily containing extended ones).

A function group (respectively, an extended function group) is a

finitely generated Kleinian group (respectively, a finitely generated

extended Kleinian group) with an invariant connected component of

its region of discontinuity.

A structural decomposition of function groups, in terms of the Klein-

Maskit combination theorems, was provided by Maskit in the middle

of the 70’s. One should expect a similar decomposition structure

for extended function groups, but it seems not to be stated in the

existing literature. The aim of this paper is to state and provide a

proof of such a decomposition structural picture.

RESUMEN

Los automorfismos conformes (respectivamente, anticonformes) de

la esfera de Riemann son dados por las transformaciones de Möbius

(respectivamente, Möbius extendidas). Un grupo Kleiniano (respec-

tivamente, un grupo Kleiniano extendido) es un grupo discreto de

transformaciones de Möbius (respectivamente, un grupo discreto de

transformaciones de Möbius y transformaciones de Möbius extendi-

das, necesariamente conteniendo extendidas).

Un grupo función (respectivamente, un grupo función extendido)

es un grupo Kleiniano finitamente generado (respectivamente, un

grupo Kleiniano extendido finitamente generado) con una compo-

nente conexa invariante de su región de discontinuidad.

Una descomposición estructural de los grupos función, en términos

de los teoremas de combinación de Klein-Maskit, fue dado por

Maskit a mediados de los 70’s. Se debiera esperar una estructura

de descomposición similar para los grupos función extendidos, pero

no parece estar enunciado en la literatura existente. El objetivo de

este art́ıculo es enunciar y dar una demostración de una tal descom-

posición estructural.

Keywords and Phrases: Kleinian groups, equivariant loop theorem.

2020 AMS Mathematics Subject Classification: 30F10, 30F40.

Accepted: 20 July, 2021

Received: 24 March, 2021

©2021 R. A. Hidalgo. This open access article is licensed under a Creative Commons

Attribution-NonCommercial 4.0 International License.

http://cubo.ufro.cl/
http://dx.doi.org/10.4067/S0719-06462021000300369
https://orcid.org/0000-0003-4070-2819


370 Rubén A. Hidalgo CUBO
23, 3 (2021)

1 Introduction

The conformal (respectively, anticonformal) automorphisms of the Riemann sphere Ĉ = C ∪ {∞}
are provided by the Möbius (respectively, extended Möbius) transformations, that is, transforma-

tions of the form T(z) = (az+b)/(cz+d) (respectively, L(z) = (az+b)/(cz+d)) where a, b, c, d ∈ C
are such that ad − bc = 1. The group of Möbius transformations M is isomorphic to the special
projective linear group PSL2(C) and the group of Möbius and extended Möbius transformations

is M̂ = ⟨M, J(z) = z⟩.

A Kleinian group (respectively, an extended Kleinian group) is a discrete subgroup of M (respec-

tively, a discrete subgroup of M̂ necessarily containing extended Möbius transformations). The

region of discontinuity of a (extended) Kleinian group K is the locus of points p ∈ Ĉ admitting
an open neighborhood p ∈ U ⊂ Ĉ such that k(U) ∩ U ̸= ∅ only for finitely many elements k ∈ K.
By definition, the region of discontinuity is an open set (it might be empty). The complement

of the region of discontinuity is called the limit set and it is the place where the dynamics of the

group action is chaotic. The history of Kleinian groups can be traced back to Poincaré [17] and a

classical source is the book [13].

A function group is a finitely generated Kleinian group (with a non-empty region of discontinu-

ity) admitting an invariant connected component of its region of discontinuity. Basic examples of

function groups are provided by elementary groups (Kleinian groups with finite limit set), quasi-

fuchsian groups (function groups whose limit set is a Jordan curve) and totally degenerate groups

(non-elementary finitely generated Kleinian groups whose region of discontinuity is both connected

and simply-connected). In a serie of papers, Maskit provided the following decomposition structure

of function groups, in terms of the Klein-Maskit combination theorems [7, 8, 13].

Theorem 1.1 (Maskit’s decomposition of function groups [6, 9, 10, 11]). Every function group

is constructed from elementary groups, quasifuchsian groups and totally degenerate groups by a

finite number of applications of the Klein-Maskit combination theorems. Moreover, in the con-

struction, the amalgamated free products and the HNN-extensions are realized along either (i) a

finite cyclic group (including the trivial group) or (ii) a cyclic group generated by an accidental

parabolic element.

An extended function group is a finitely generated extended Kleinian group with an invariant

connected component of its region of discontinuity. Basic examples of extended function groups

are the extended elementary groups (extended Kleinian groups with finite limit set), extended

quasifuchsian groups (finitely generated extended function groups whose limit set is a Jordan

curve) and extended totally degenerate groups (non-elementary extended finitely generated Kleinian

groups with connected and simply-connected region of discontinuity).

Note that the term “extended quasifuchsian group” used in this paper is different from the given


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The structure of extended function groups 371

by other authors in the sense that they refer it to Kleinian groups whose limit set is a Jordan curve

and contains elements permuting the two discs bounded by it.

As it is for the case of function groups, one should expect a similar decomposition result for the

extended function groups (see Theorem 1.2). It seems that such a result is missing in the literature.

The aim of this paper is to provide such an structural decomposition of extended function groups.

Theorem 1.2 (Decomposition of extended function groups). Every extended function group is

constructed from (extended) elementary groups, (extended) quasifuchsian groups and (extended)

totally degenerate groups by a finite number of applications of the Klein-Maskit combination theo-

rems. Moreover, in the construction, the amalgamated free products and the HNN-extensions are

realized along either (i) a finite cyclic group (including the trivial group) or (ii) an infinite dihedral

group generated by two reflections or (iii) a cyclic group generated by either an accidental parabolic

element or by an accidental pseudo-parabolic element (i.e., its square is accidental parabolic).

The above structure description is a consequence of Theorems 3.1 and 3.2 (which are new results);

their statements are at the beginning of section 3. Their proofs build upon a sequence of lemmas

3.5, 3.7, 3.8, 3.9, which also are not found in the literature.

The idea of the proof is the following. Let K be an extended function group, with invariant

connected component ∆. Its index two orientation preserving half K+ = K ∩ M is a function
group with the same invariant component. As K+ is finitely generated, Selberg’s lemma [19]

asserts the existence of a torsion free finite index normal subgroup G1 of K
+ (which is again a

function group). Since K = ⟨K+, τ⟩, where τ2 ∈ K+, the group G = G1 ∩τG1τ−1 is a finite index
torsion free normal subgroup of K. By Ahlfors finiteness theorem [1], the quotient space S = ∆/G

is an analytically finite Riemann surface, that is, S = Ŝ − C, where Ŝ is a closed Riemann surface
and C ⊂ Ŝ is a finite set of points (it might be empty). The finite group H = K/G is a group of
conformal and anticonformal automorphisms of S. Maskit’s decomposition of function groups may

be applied to G. There are many possible decompositions, but in order to get one which can be used

to obtain a decomposition of K, we must find one which is in some sense equivariant with respect

to G. This is solved by Theorem 2.2 (equivariant theorem for function groups) obtained by Maskit

and the author in [4]. This result permits us to obtain a first decomposition structural picture (see

Theorem 3.1). In such a picture, there may appear (extended) B-groups as factors. A B-group

(respectively, an extended B-group) is a function group (respectively, an extended function group)

with a simply-connected invariant component in its region of discontinuity. A subtle modification

to Maskit’s arguments for the case of B-groups, to deal with these extended B-groups, is provided

(see Theorem 3.2).

A. Haas’s thesis [3] concerns with uniformizing groups of conformal and anticonformal automor-

phisms acting on plane domains. It leads naturally to extended function groups, but it seems that

the above decomposition does not follow immediately from it.


372 Rubén A. Hidalgo CUBO
23, 3 (2021)

2 Preliminaries

2.1 Riemann orbifolds

A Riemann orbifold O consists of a (possible non-connected) Riemann surface S (called the under-
lying Riemann surface of the orbifold), an isolated collection of points of S (called the cone points

of the orbifold) and associated to each cone point an integer at least 2 (called the cone order). A

connected Riemann orbifold is analytically finite if its underlying (connected) Riemann surface is

the complement of a finite number of points of a closed Riemann surface and the number of cone

points is also finite. We may think of a Riemann surface as a Riemann orbifold without cone points.

A conformal automorphism (respectively, anticonformal automorphism) of the Riemann orbifold

O is a conformal automorphism (respectively, anticonformal) of the underlying Riemann surface S
which preserves both its set of cone points together with their cone orders (cone points can be per-

muted but preserving their cone orders). We denote by Aut(O) (respectively, Aut(S)) the group
of conformal/anticonformal automorphisms of O (respectively, S) and by Aut+(O) (respectively,
Aut+(S)) its subgroup of conformal automorphisms.

2.2 Kleinian and extended Kleinian groups

In the following, we recall some facts on (extended) Kleinian groups. A good source on the topic

are the classical books [13, 14]. Let us start by observing that, if K1 < K2 < M̂ and K1 has finite

index in K2, then both are discrete if one of them is and, in the discreteness case, both have the

same region of discontinuity.

Let K < M̂ and set K+ := K ∩ M. If K ̸= K+, then K+ is called the orientation-preserving half
of K and, in this case, K is an extended Kleinian group if and only if K+ is a Kleinian group;

in which case both have the same region of discontinuity. If moreover, K is an extended Kleinian

group and K+ is a function group, then either: (i) K is an extended function group or (ii) K+ is

a quasifuchsian group and there is an element of K − K+ permuting both discs bounded by the
limits set Jordan curve (so K is not an extended function group).

2.3 Accidental parabolic elements

A B-group is a function group K with a simply-connected invariant component ∆. Let us assume

K is non-elementary (i.e., its limit set is not finite). By the Klein-Poincaré uniformization theorem

[18], there is a bi-holomorphism f : H2 → ∆, where H2 denotes the hyperbolic upper-half plane.
The group Γ = f−1Kf is a group of conformal automorphisms of H2, i.e., a fuchsian group of the

first kind, in particular, a B-group with H2 as an invariant connected component of its region of

discontinuity. In this case, H2/Γ has finite hyperbolic area. It is known that f sends parabolic


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The structure of extended function groups 373

transformations to parabolic transformations, but it may send a hyperbolic transformation to a

parabolic one. A parabolic element P ∈ K is called accidental if f−1Pf is a hyperbolic transfor-
mation. In this case, the image under f of the axis of the hyperbolic transformation f−1Pf is

called the axis of P (in Maskit’s notation this is the true axis of P).

If K is an extended B-group, that is, an extended function group with a simply-connected invariant

component, then we say that an element of K is accidental pseudo-parabolic if its square is an

accidental parabolic element of K+.

2.4 Klein-Maskit’s decomposition theorems

Let K be a Kleinian group with region of discontinuity Ω and let H be a subgroup of K with limit

set Λ(H). A set X ⊂ Ĉ is called precisely invariant under H in K if E(X) = X, for every E ∈ H,
and T(X) ∩ X = ∅, for every T ∈ K \ H.

We will assume H to be either (i) the trivial group, (ii) a finite cyclic group or (iii) an infinite cyclic

group generated by a parabolic transformation. If H is a cyclic subgroup, a precisely invariant

disc B is the interior of a closed topological disc B, where B − Λ(H) ⊂ Ω is precisely invariant
under H in K.

Theorem 2.1 (Klein-Maskit’s combination theorems [7, 8]).

(1) (Amalgamated free products). For j = 1, 2, let Kj be a Kleinian group, let H ≤ K1 ∩ K2 be
a cyclic subgroup (either trivial, finite or generated by a parabolic transformation), H ̸= Kj, and
let Bj be a precisely invariant disc under H in Kj. Assume that B1 and B2 have as a common

boundary the simple loop Σ and that B1 ∩ B2 = ∅. Then K = ⟨K1, K2⟩ is a Kleinian group
isomorphic to the free product of K1 and K2 amalgamated over H, that is, K = K1 ∗H K2, and
every elliptic or parabolic element of K is conjugated in K to an element of either K1 or K2.

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

(2) (HNN extensions). Let K be a Kleinian group. For j = 1, 2, let Bj be a precisely invariant

disc under the cyclic subgroup Hj (either trivial, finite or generated by a parabolic) in K, let Σj be

the boundary loop of Bj and assume that T(B1)∩B2 = ∅, for every T ∈ K. Let A be a loxodromic
transformation such that A(Σ1) = Σ2, A(B1) ∩ B2 = ∅, and A−1H2A = H1. Then KA = ⟨K, A⟩
is a Kleinian group, isomorphic to the HNN-extension K∗⟨A⟩ (that is, every relation in KA is
consequence of the realtions in K and the relations A−1H2A = H1). If each Hj, for j = 1, 2, is

its own normalization in K, then every elliptic or parabolic element of KA is conjugated to some

element of K. Moreover, if K is geometrically finite, then KA is also geometrically finite.


374 Rubén A. Hidalgo CUBO
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2.5 An equivariant loop theorem for function groups

Let K be a function group and ∆ be a K-invariant connected component of its region of dis-

continuity. By the Alhfors’ finiteness theorem [1, 2], the quotient O = ∆/K turns out to be an
analytically finite Riemann orbifold. Let B ⊂ O be the (finite) collection of the cone points and
let G ⊂ O − B be the collection of loops which lift to loops under the natural regular holomorphic
covering π : ∆0 → O − B, where ∆0 is the open dense subset of ∆ consisting of those points with
trivial K-stabilizer. In [5], Maskit proved the existence of a finite subcollection F ⊂ G of pairwise
disjoint loops inside O − B, each one being a finite power of a simple loop, such that the cover
π is determined as a highest regular planar cover for which the loops in F lift to loops (such a
collection of loops is not unique). The collection F is called a fundamental system of loops of the
above regular planar covering. Assume that there is a finite group H < Aut(O) whose elements
lift to automorphisms of ∆ under π. Then, in [4], Maskit and the author proved that there is a

fundamental system of loops F being equivariant under H.

Theorem 2.2 (Equivariant loop theorem for function groups [4]). Let K be a function group, with

invariant connected component ∆ in its region of discontinuity, O = ∆/K (which is an analytically
finite Riemann orbifold) and let B be the finite set of cone points of O. Let π : ∆ → O be the
natural regular branched regular covering induced by K. Let G be the collection of loops in O − B
which lift to loops in ∆ under π. If H < Aut(O) lifts to a group of automorphisms of ∆, then
there is a finite sub-collection F ⊂ G such that:

(1) F consists of pairwise disjoint powers of simple loops;

(2) F is H-invariant; and

(3) every loop in G is homotopic to the product of finite powers of a finite loop in F.

The collection F is called a fundamental set of loops for the pair (K, H).

Remark 2.3. The condition (3) above is equivalent to say that F is a fundamental system of
loops for π. Also, if the function group K is torsion-free, then O is an analytically finite Riemann
surface and each of the loops in the finite collection F turns out to be a simple loop.

As a consequence of the above, one may write the following equivariant result for Kleinian groups.

Theorem 2.4 (Equivariant loop theorem for Kleinian groups). Let K be a Kleinian group with

region of discontinuity Ω ̸= ∅, let ∆ be a (non-empty) collection of connected components of Ω
which is invariant under the action of K, let O = ∆/K, let B be the cone points of O and let
H < Aut(O) be a finite group of automorphisms of O. Let us assume that O consists of (may be
infinitely many) analytically finite Riemann orbifolds. Fix some regular (branched) covering map

π : ∆ → O with K as its deck group. Let G be the collection of loops in O−B which lift, with respect


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The structure of extended function groups 375

to π, to loops in ∆. If H lifts to a group of automorphisms of ∆, then there is a sub-collection

F ⊂ G such that:

(1) F consists of pairwise disjoint powers of simple loops;

(2) F is H-invariant; and

(3) every loop in G is homotopic to the product of finite powers of a finite sub-collection of loops
in F.

Proof. Let us consider a maximal subcollection of non-equivalent components of ∆ under the action

of K, say ∆j for j ∈ J. Let Kj be the K-stabilizer of ∆j under the action of K. By Theorem 2.2,
on Oj = ∆j/Kj there is a collection of loops, say Fj, satisfying the properties on that theorem.
Clearly the collection of fundamental loops F = ∪j∈JFj is the required one.

Remark 2.5. The condition for O = ∆/K to consist of analytically finite Riemann orbifolds is
equivalent, by the Ahlfors finiteness theorem, for the K-stabilizer of each connected component in

∆ to be finitely generated. In particular, if K is finitely generated, then O is a finite collection of
analytically finite Riemann surfaces and F turns out to be a finite collection. If, in Theorem 2.4,
we assume K to be torsion-free, then the loops in F will be simple loops.

2.6 A connection to Kleinian 3-manifolds

Let K be a Kleinian group, with region of discontinuity Ω ⊂ Ĉ. There is a natural discrete action
(by Poincaré extension) of K on the upper half-space H3 = {(z, t) : z ∈ C, t ∈ (0, +∞)}, which is
given by isometries in the hyperbolic metric ds2 = (|dz|2+dt2)/t2. The quotient MK = (H3∪Ω)/K
carries the structure of a 3-orbifold, its interior H3/K has a structure of a complete hyperbolic

3-orbifold and Ω/K the structure of a Riemann orbifold. In the case that K is torsion free, all the

above are manifolds and we say that MK is a Kleinian 3-manifold.

A direct consequence of Theorem 2.4 is the equivariant theorem for Kleinian 3-manifolds in the case

that the conformal boundary is non-empty and it consists of analytically finite Riemann surfaces.

Corollary 2.6. Let K be a torsion free Kleinian group, with non-empty region of discontinuity

Ω, such that SK = Ω/K is a collection (it might be infinitely many of them) of analytically

finite Riemann surfaces. Let H be a finite group of automorphismsm of the Kleinian 3-manifold

MK = (H3 ∪ Ω)/K. If G is the collection of loops on SK that are homotopically nontrivial in
SK but homotopically trivial in MK, then there exists a collection of pairwise disjoint simple loops

F ⊂ G, equivariant under the action of H, so that G is the smallest normal subgroup of π1(SK)
generated by F.


376 Rubén A. Hidalgo CUBO
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Remark 2.7. Let K be a torsion free Kleinian group and let H be as in Corollary 2.6. Then

the following hold. (1) If π1(M) is finitely generated, then the collection F is finite. (2) By
lifting H to the universal cover space, one obtains a (extended) Kleinian group K̂ containing K

as a finite index normal subgroup so that H = K̂/K. Corollary 2.6 may be used to obtain a

geometric structure picture of K̂, in the sense of the Klein-Maskit combination theorems, in terms

of the algebraic structure of H. (3) If MK is compact, then the result follows from Meeks-Yau’s

equivariant loop theorem [15, 16], whose arguments are based on minimal surfaces theory. If K

is not a purely loxodromic geometrically finite Kleinian group, then MK is non-compact and the

result is no longer a consequence of Meek’s-Yau’s equivariant theorem.

3 Proof of Theorem 1.2

The proof of Theorem 1.2 is a direct consequence of Theorem 3.1, which is the main step, and

Theorem 3.2 as described below. If the word “extended” is removed, the statements of these

theorems are simply Maskit’s original theorems (see [6, 9, 10, 11]).

Theorem 3.1 (First step in Maskit-type decomposition of an extended function group). Every

extended function group is constructed, using the Klein-Maskit combination theorems, as amalga-

mated free products and HNN-extensions using a finite collection of (extended) B-groups. Moreover,

the amalgamations and HNN-extensions are realized along either trivial or a finite cyclic group or

a dihedral group generated by two reflections (this last one only in the amalgamated free product

operation).

The above result asserts that every extended function group is constructed from (extended) B-

groups by applying the Klein-Maskit combination theorems. Maskit’s results provide a geometrical

decomposition of B-groups (see Theorem 3.2 below and delete the word “extended”). We now need

to take care of the extended B-groups, which is exactly what the next result is about.

Theorem 3.2 (Decomposition of extended B-groups). Let K be an extended B-group with a

simply-connected invariant component ∆. Then either (i) K is an elementary extended Kleinian

group or (ii) K is an extended quasifuchsian group or (iii) K is an extended degenerate group or (iv)

∆ is the only invariant component and K is constructed as amalgamated free products and HNN-

extensions, by use of the Klein-Maskit combination theorems, using (extended) elementary groups,

(extended) quasifuchsian groups and (extended) totally degenerate groups. The amalgamated free

products and HNN-extensions are given along axes of accidental parabolic transformations.

Remark 3.3. We note for the reader that the proof of Theorem 3.1 includes Remarks 3.4 and 3.6

and Lemmas 3.5 and 3.7 and that the proof of Theorem 3.2 includes Lemmas 3.8 and 3.9.


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The structure of extended function groups 377

3.1 Proof of Theorem 3.1

Let K be an extended function group and let ∆ be a K-invariant connected component of its

region of discontinuity (we may assume K to be non-elementary). If there is another different

invariant connected component of its region of discontinuity, then K+ = K ∩ M ̸= K is known
to be a quasifuchsian group [12]; so K is an extended quasifuchsian group. Let us assume, from

now on, that ∆ is the unique invariant connected component. By Selberg’s lemma [19], there is a

torsion free finite index normal subgroup G1 of K
+. As K = ⟨K+, τ⟩, where τ2 ∈ K+, one has

that G = G1 ∩ τG1τ−1 is a torsion free finite index normal subgroup of K.

It follows that G is a function group with ∆ as an invariant connected component of its region

of discontinuity (the same as for K). Also, ∆ is the only invariant connected component of G;

otherwise G is a quasifuchsian group and K will have two different invariant connected components,

which is a contradiction to our assumption on K.

Let S = ∆/G (an analytically finite Riemann surface by Ahlfors finiteness theorem) and consider

a regular planar unbranched cover P : ∆ → S with G as its deck group. Set H = K/G <
Aut(S), which is a non-trivial finite group (since G ̸= K). Theorem 2.2 asserts the existence of a
fundamental set of loops F ⊂ S for the pair (G, H). Such a collection of loops cuts S into some
finite number of connected components and such a collection of components is invariant under

H. The H-stabilizer of each of these connected components and each of the loops in F is a finite
group.

Remark 3.4 (Decomposition structure of H). The H-equivariant fundamental system of loops F
permits to obtain a structure of H as a finite iteration of amalgamated free products and HNN-

extensions of certain subgroups of H as follows. Let us consider a maximal collection of components

of S −F, say S1. . . , Sn, so that any two different components are not H-equivalent. Let us denote
by Hj the H-stabilizer of Sj. It is possible to choose these surfaces so that, by adding some on the

boundary loops, we obtain a planar surface S∗ (containing each Sj in its interior). If two surfaces

Si and Sj have a common boundary in S
∗, then Hi ∩ Hj is either trivial or a cyclic group (this

being exactly the H-stabilizer of the common boundary loop). We perform the amalgamated free

product of Hi and Hj along the trivial or cyclic group Hi ∩ Hj. Set Sij be the union of Si, Sj
with the common boundary loop in S∗ and set Hij the constructed group. Now, if Sk is another

of the surfaces which has a common boundary loop in S∗ with Sij, then we again perform the

amalgamated free product of Hij and Hk along the trivial or cyclic group Hij ∩ Hk. Continuing
with this procedure, we end with a group H∗ obtained as amalgamated free product along finite

cyclic groups or trivial groups. For each boundary of S∗ we add a boundary loop, in order to stay

with a planar compact surface (we are out of S in this part). If α is any of the boundary loops

of S∗, there should be another boundary loop β of S∗ and an element h ∈ H so that h(α) = β.
By the choice of the surfaces Sj, we must have that h(S

∗) ∩ S∗ = ∅. In particular, β ̸= α. If


378 Rubén A. Hidalgo CUBO
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there is another element k ∈ H − {h} so that k(α) = β, then k−1h is a non-trivial element that
stabilizes α and k−1h(S∗) ∩ S∗ ̸= ∅, a contradiction. Also, if there is another boundary loop γ of
S∗ (different from β) and an element u ∈ H so that u(α) = γ, then uh−1 ∈ H − {I} satisfies that
uh−1(S∗) ∩ S∗ ̸= ∅, which is again a contradiction. We may now perform the HHN-extension of
H∗ by the finite cyclic group generated by h. If α1 = α,. . . , αm are the boundary loops of S

∗,

which are not H-equivalent, then we perform the HHN-extension with each of them. At the end,

we obtain an isomorphic copy of H.

We may assume the fundamental set of loops F to be minimal, that is, by deleting any non-empty
subcollection of loops from it, we obtain a collection which fails to be a fundamental set of loops

for (G, H). The minimality condition asserts that each connected component of S − F is different
from either a disc or an annulus. By lifting F to ∆, under P , one obtains a collection F̂ ⊂ ∆ of
pairwise disjoint simple loops, so that F̂ is invariant under the group K. Each of the loops in F̂ is
called a structure loop and each of the connected components of ∆ − F̂ a structure region. These
structure loops and regions are permuted by the action of K. The K-stabilizer (respectively, the

G-stabilizer) of each structure loop and each structure region is called a structure subgroup of K

(respectively, a structure subgroup of G).

If R is a structure region, then its K-stabilizer, denoted by KR, is a finite extension of its G-

stabilizer, denoted by GR. Similarly, if α is a structure loop, then its K-stabilizer is a finite

extension of its G-stabilizer.

Lemma 3.5. Let α be a structure loop and let R be a structure region containing α on its border.

Then the KR-stabilizer of α is either trivial or a finite cyclic group or a dihedral group generated by

two reflections (both circles of fixed points intersecting at two points, one inside of one of the two

discs bounded by α and the other point contained inside the other disc). Moreover, the K-stabilizer

of α is either equal to its KR-stabilizer or it is generated by its KR-stabilizer and an involution

(conformal or anticonformal) that sends R to the other structure region containing α in its border.

Proof. Let α ∈ F̂ be a structure loop. As α is contained in the region of discontinuity of K, the
K-stabilizer of α is a finite group; so also its G-stabilizer is finite. Note that the K+-stabilizer of

α is either trivial, a finite cyclic group or a dihedral group. Moreover, in the dihedral case, one of

the involutions interchanges both discs bounded by α. Let R be a structure region containing α

as a boundary loop. Then the K+R -stabilizer of α is either trivial or a finite cyclic group. It follows

that the KR-stabilizer of α is either trivial or a finite cyclic group or a dihedral group generated

by two reflections (both circles of fixed points intersecting at two points, one inside of one of the

two discs bounded by α and the other point contained inside the other disc). The K-stabilizer of

α is generated by the KR-stabilizer and probably an extra involution (conformal or anticonformal)

that interchanges both discs bounded by α.


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The structure of extended function groups 379

Remark 3.6. We do not need this extra information for the rest of the proof, but it may help with

a clarification of the gluing process at the Klein-Maskit combination theorems. It follows, from

Lemma 3.5, that the K-stabilizer of α ∈ F̂ must be one of the following: (1) the trivial group, (2)
a cyclic group generated by a reflection with α as its circle of fixed points (so it permutes both discs

bounded by α), (3) a cyclic group generated by a reflection that keeps invariant each of the two

discs bounded by α (the reflection has exactly two fixed points over α), (4) a cyclic group generated

by an imaginary reflection (it permutes both discs bounded by α), (5) a cyclic group generated by

an elliptic transformation of order two (permuting the two discs bounded by α), (6) a cyclic group

generated by an elliptic transformation (preserving each of the two discs bounded by α), (7) a

group generated by an elliptic transformation (preserving each of the two discs bounded by α) and

a reflection whose circle of fixed points is α, (8) a group generated by an elliptic transformation

(preserving each of the two discs bounded by α) and an imaginary reflection (permuting both discs

bounded by α), (9) a group generated by an elliptic transformation of order two (permuting the

two discs bounded by α) and an imaginary reflection that keeps α invariant (it permutes both

discs bounded by α), (10) a dihedral group generated by two reflections (both circles of fixed points

intersecting at two points, one inside of one of the two discs bounded by α and the other point

contained inside the other disc), (11) a group generated by an elliptic transformation (preserving

each of the two discs bounded by α) and an imaginary reflection that keeps α invariant (it permutes

both discs bounded by α), (12) a group generated by a dihedral group of Möbius transformations and

a reflection with α as circle of fixed points, (13) a group generated by a dihedral group generated

by two reflections (both circles of fixed points intersecting at two points, one inside of one of

the two discs bounded by α and the other point contained inside the other disc) and an elliptic

transformation of order two that permutes both discs bounded by α, To obtain the above, we use

the following fact. Let α be a loop which is invariant under (i) an elliptic transformation E, of

order two that interchanges both discs bounded by it, and (ii) also invariant under an imaginary

reflection τ. Then Eτ is necessarily a reflection whose circle of fixed points is transversal to α.

Let R be a structure region and let α ∈ F̂ be on the boundary of R. By Lemma 3.5, the KR-
stabilizer of α is some finite group; either trivial or a finite cyclic group or a dihedral group

generated by two reflections (both circles of fixed points intersecting at two points, one inside of

one of the two discs bounded by α and the other point contained inside the other disc). Let Dα be

the topological disc bounded by α and disjoint from R. Clearly, the KR-stabilizer of such a disc is

contained in the KR-stabilizer of α (each element of KR that stabilizes Dα also stabilizes α), so

Dα is contained in the region of discontinuity of KR. It follows that KR is a (extended) function

group with an invariant connected component ∆R of its region of discontinuity containing R and

all the discs Dα, for every structure loop α on its boundary.


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Lemma 3.7. ∆R is simply-connected.

Proof. If ∆R is not simply-connected, then there is a simple loop β ⊂ R bounding two topological
discs, each one containing limit points of KR (so limit points of K). The projection on S of β

produces a loop β̃ ⊂ S which lifts to a loop under P . But, we know that β̃ is homotopic to the
product of finite powers of the simple loops on the boundary of the finite domain P(R) ⊂ S. It
follows that β must be homotopic to the product of finite powers of a finite collection of structure

loops on the boundary of R. As each of these boundary loops bounds a disc containing no limit

points, we get a contradiction for β to bound two discs, each one containing limit points.

We may follow the same lines as described in Remark 3.4 to obtain that K is constructed, using

the Klein-Maskit combination theorems [13, 7], as amalgamated free products and HNN-extensions

using a finite collection of the structure subgroups of KR (which, by Lemma 3.7, are extended

B-groups with invariant simply-connected component ∆R). By Lemma 3.5, the amalgamations

and HNN-extensions are realized along either trivial or a finite cyclic group or a dihedral group

generated by two reflections. This ends the proof of Theorem 3.1. □

3.2 Proof of Theorem 3.2

We proceed to describe the subtle modifications in Maskit’s arguments in the decomposition of

B-groups [10, 11] adapted to the case of extended B-groups (see also chapter IX.H. in [13]). Let us

assume that K is an extended B-group and that it is neither an (extended) elementary group or a

(extended) quasifuchsian group or a (extended) degenerate group. Let ∆ be the simply-connected

invariant component of the region of discontinuity of K. Every other connected component of

the region of discontinuity of K is simply-connected (see Proposition IX.D.2. in [13]). By our

assumptions on K, we have that K+ is neither elementary nor degenerate Kleinian group. It may

be, even if K is not an extended quasifuchsian, that K+ is a quasifuchsian. But in this case, we

have that K is just a HNN-extension of a quasifuchsian group along a cyclic group. So, from now

on, we assume that K+ is neither a quasifuchsian group.

As K is non-elementary, we may consider a bi-holomorphism f : H2 → ∆ and consider the fuch-
sian group f−1Kf. As it is well known that no rank two parabolic subgroup can preserve a disc

in the Riemann sphere, it follows that f−1K+f does not contain rank two parabolic subgroup,

in particular, K+ neither does contain a rank two parabolic subgroup. Theorem IX.D.21 in [13]

states that K+ is either quasifuchsian or totally degenerate or it contains accidental parabolics.

By our assumptions on K and K+, we note that K+ necessarily must have accidental parabolic

transformations. Moreover, there is a finite number of conjugacy classes of primitive accidental

parabolic transformations in K+. Let us consider a collection of accidental parabolic transforma-

tions in K+, say P1,..., Pm, so that Pj is not K
+-conjugate to P ±1r if j ̸= r, and Pj is primitive,


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The structure of extended function groups 381

that is it is not of the form Qa for some Q ∈ K and a ≥ 2. Let us denote by Lj ⊂ ∆ the axis of
Pj (note that Lj is a geodesic for the hyperbolic metric of ∆ and that Pj keeps it invariant acting

by a translation on it).

Lemma 3.8. (1) If j ̸= r, then the K+-translates of Lj do not intersect the K+-translates of Lr.
(2) For each fixed j, any K+-translates of Lj is either disjoint from Lj or it coincides with it.

Proof. Let us consider a Riemann map f : H2 → ∆, where H2 is the upper half-plane with the
hyperbolic metric ds2 = |dz|2/Im(z)2. It is well known that any two different geodesics in H2 are
either disjoint of they intersect at exactly one point. The push-forward of the hyperbolic metric

in H2 provides the hyperbolic metric of ∆. It follows that any K+-translate of Lj and any K+

translate of Lr (for j not necessarily different from r) are either disjoint or they intersect exactly

at one point or they are the same. Let us first prove (1), that is, we assume j ̸= r. If there are
K+-translates of Lj and Lr which are the same, as Pj and Pr are primitive parabolic, share the

same fixed point and K+ is discrete, then Pj is conjugate to either P
±1
r , a contradiction. If there

are K+-translates of Lj and Lr which intersect at a point, then the planarity of ∆ asserts that

the non-empty intersection only may happen if a K+ conjugate of Pj and a K
+-conjugate of Pr

share their unique fixed point. The discreteness of K+ asserts that K+ must contain a rank two

parabolic subgroup, a contradiction. Let us now prove (2), that is, we assume j = r. This follows

the same lines a the previous case to see that either the translates are either disjoint or equal.

Lemma 3.9. If T ∈ K − K+, then T preserves the collection of K+-translates of {L1, ...., Lm}.

Proof. T acts as an isometry on ∆ and must permute the accidental parabolic transformations.

As the axis is unique for each accidental parabolic, we are done.

Let L̂j be equal to Lj together the corresponding fixed point of Pj. Then the collection F given
by the K+-translates of {L̂1, ..., L̂m} consists of pairwise disjoint simple loops; each one is called
a structure loop for the group K. Such a collection of structure loops is still invariant for any

T ∈ K − K+ by Lemma 3.9. The structure loops cut Ω (the region of discontinuity of K) and ∆
into regions; called structure regions for K. These are different from our previous definitions of

structure loops and regions as these ones are not completely contained in the region of discontinuity.

Let α ∈ F be a structure loop and let R1 and R2 be the two structure regions containing α in
their common boundary. Let Kj < K be the K-stabilizer of Rj, let Kα be the K-stabilizer of

α and let P ∈ K be the primitive accidental parabolic transformation whose axis is α (which is
then K-conjugated to some of the Pj’s). Clearly, ⟨P⟩ is contained in Kj, ⟨P⟩ < Kα and either
(i) ⟨P⟩ = Kα or (ii) ⟨P⟩ has index two in Kα or (iii) ⟨P⟩ has index four in Kα (this last case
means that ⟨P⟩ has index two inside the Kj-stabilizer of α). The region R3−j is contained in
a disc D3−j, whose Kj-stabilizer is equal to the Kj-stabilizer of the loop α; this is either the


382 Rubén A. Hidalgo CUBO
23, 3 (2021)

cyclic group generated by P or it contains it as an index two subgroup. It follows that D3−j

is contained in the region of discontinuity Ωj of Kj and that there is an invariant connected

component ∆j ⊂ Ωj containing ∆. Lemma IX.H.10 in [13] states that K+j is a B-group, with
∆j as invariant simply-connected component, without accidental parabolic transformations. It

follows that K+j is either elementary or quasifuchsian or totally degenerate, in particular, that Kj

is either (extended) elementary or (extended) quasifuchsian or (extended) totally degenerate. One

possibility is that Kα is an extension of degree two of the Kj-stabilizer of α. In this case, there is

an element Q ∈ Kα that permutes R1 with R2 (Q is either a pseudo-parabolic whose square is P or
an involution). In this case, ⟨K1, K2⟩ is the HNN-extension of K1 by Q (in the sense of the second
Klein-Maskit combination theorem). The other possibility is that Kα is equal to K1 ∩ K2 (either
the cyclic group generated by the parabolic P or a group generated by two reflections sharing as

a common fixed point the fixed point of P). In this case, ⟨K1, K2⟩ is the free product of K1 and
K2 amalgamated over K1 ∩ K2 (in the sense of the first Klein-Maskit combination theorem).

Now, following the same ideas in [10, 11], one obtains a decomposition of K as an amalgamated free

products and HNN-extensions, by use of the Klein-Maskit combination theorems, using (extended)

elementary groups, (extended) quasifuchsian groups and (extended) totally degenerate groups. □

Acknowledgment

The author would like to thank the referees for their valuable comments, suggestions and corrections

to the previous versions.


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The structure of extended function groups 383

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	Introduction
	Preliminaries
	Riemann orbifolds
	Kleinian and extended Kleinian groups
	Accidental parabolic elements
	Klein-Maskit's decomposition theorems
	An equivariant loop theorem for function groups
	A connection to Kleinian 3-manifolds

	Proof of Theorem 1.2
	Proof of Theorem 3.1
	Proof of Theorem 3.2