Journal of Lie Theory Volume 11 (2001) 491–503 \begin{matrix}c \\ ◯ \end{matrix} 2001 Heldermann Verlag

Some Examples of Discrete Groups and Hyperbolic Orbifolds of Infinite Volume

Elena Klimenko^{∗}

Communicated by E. B. Vinberg

Abstract. We study the class of two-generator subgroups of PSL(2, C) with real parameters which was introduced recently by Gehring, Gilman, and Martin. We give criteria for discreteness of non-elementary and non-Fuchsian groups of this class that are generated by two hyperbolic elements. We construct all hyperbolic orbifolds uniformized by the discrete groups of such type. The orbifolds described are of infinite volume.

Introduction

Throughout this paper we will consider PSL(2, C) which acts as the full group of orientation preserving isometries of hyperbolic space H^{3}. In the last years its 2-generator subgroups have been the subject of investigation from different points of view, see [1], [3], [4], [8], [9], [12]–[20], [26], [27] (this list is not complete). A series of papers is devoted to the following problem:

Problem. When is ⟨f, g⟩ < PSL(2, C) discrete?

By now, only necessary or only sufficient conditions for the discreteness are presented in most of the papers. For the former we mention the remarkable Jørgensen inequality, the inequality of Shimizu–Leutbecher and their analogs, see eg. [15]–[16]. For the latter see [9] and references therein.

Given a discrete subgroup Γ of PSL(2, C), one can consider the orbit space H^{3}/Γ, which is a hyperbolic orbifold. Therefore, we can regard different examples of 2-generator hyperbolic orbifolds as a kind of sufficient conditions for discreteness. Such examples were constructed, for instance, in [12]–[14].

Still there are no criteria (that is, necessary and sufficient conditions) to determine the discreteness of 2-generator groups. It is quite natural to solve this problem first for particular classes of subgroups of PSL(2, C). What are the classes completely described?

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag

∗ The research was partially supported by Sonderforschungsbereich 343 “Diskrete Strukturen in der Mathematik” (Universität Bielefeld) and by the Russian Foundation for Basic Research (Grant 99–01–00630)

The description of all elementary discrete groups is well known (see [2], [7], or [25]). If both generators keep invariant a hyperbolic plane in H^{3}, then the problem was also completely solved: if the generators preserve orientation of that plane, see [7], [10], [11], and [28]; if at least one of the generators reverses orientation of the invariant plane, see [23]. From the geometric point of view, almost all 2-generator discrete groups in this paragraph (except some elementary groups) are, in a sense, 2-dimensional (all of them are isomorphic to discrete subgroups of E^{2}, S^{2}, or H^{2}, in the last case they are not necessarily orientationpreserving).

In [8], Gehring, Gilman, and Martin suggest the investigation of the class of two-generator groups with real parameters (β, β′, γ), where β = tr^{2}f − 4, β′ = tr^{2}g − 4, and γ = tr[f, g] − 2. In case when such a group is discrete they obtained necessary conditions on the parameters. In [20], a geometric characterization of real parameters is given, moreover, it is shown that the class contains many “truly spatial” groups (non-elementary groups without invariant plane). We study those spatial groups in series of papers [17]–[22].

The purpose of the present paper is to describe all non-Fuchsian groups with real parameters generated by two hyperbolic elements. Namely, we find necessary and sufficient conditions for their discreteness and construct the hyperbolic orbifolds uniformized by the discrete groups of such type. The orbifolds described are of infinite volume.

Acknowledgements. Part of this work was done while the author was staying at the University of Bielefeld. I express my sincere gratitude to Prof. H. Helling, Prof. A. Mednykh, and Prof. G. Rosenberger for very useful discussions. I would like to thank N. Kopteva for the illustrations and Prof. B. Kunyavski for numerous remarks.

1. Preliminaries

1.1. Hyperbolic geometry.

We recall some facts of hyperbolic geometry.

Two (hyperbolic) planes are either intersecting, or parallel, or disjoint. Two (hyperbolic) lines either lie in a plane (and then they are intersecting, parallel, or disjoint), or they are skew.

In the next section we will need the following three propositions.

Proposition 1.1. Let ε and ρ be mutually orthogonal planes, and let a be the line by which they intersect. Let b be a straight line which lies in ρ and is orthogonal to a. Then b is orthogonal to ε.

Proposition 1.2. If a plane ε is orthogonal to a straight line b lying in another hyperbolic plane σ, then ε and σ are mutually orthogonal.

Propositions 1.1 and 1.2 are simple and occur not only in hyperbolic geometry, but also in Euclidean and spherical ones.

Proposition 1.3. Let σ_{1} and σ_{2} be disjoint planes in H^{3}, and let ε be a plane which is orthogonal to both of them. Then ε passes through the common perpendicular to σ_{1} and σ_{2}.

Proof. Indeed, the lines of intersection of σ_{1} and σ_{2} with ε are disjoint and, consequently, have their common perpendicular (lying in ε). That perpendicular is orthogonal to both σ_{1} and σ_{2} by Proposition 1.1. The proof is complete.

We recall also that the angle between two skew lines in H^{3} is defined to be the dihedral angle between two planes which pass through the common perpendicular to these lines and each of them separately. Two lines are mutually orthogonal if and only if there is a plane containing one of the lines and orthogonal to the other.

1.2. PSL(2, C) as the isometry group of hyperbolic 3-space.

Consider 3-dimensional Lobachevsky space realized as the upper half-space

H^{3} = { (z, t) | z ∈ C, t > 0}

with the hyperbolic metric

ds^{2} = (|dz|^{2} + dt^{2})/t^{2}.

As is known, PSL(2, C) acts as the full group of orientation preserving isometries of hyperbolic space H^{3}. A Kleinian group is any discrete subgroup of PSL(2, C).

We need a classification of elements of PSL(2, C). For f ∈ PSL(2, C),

let ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, C) be one of the two matrices which represent f. Then as

usual tr^{2}f ≡ (a + d)^{2}. A non-trivial element f is elliptic (parabolic, hyperbolic, or strictly loxodromic) if and only if tr^{2}f ∈ [0, 4) (tr^{2}f = 4, tr^{2}f ∈ (4, +∞), or tr^{2}f ∉ [0, +∞), respectively). Geometrically this is rotation (respectively, limit rotation, translation or rotation followed by translation with the same axis).

An elliptic, parabolic or hyperbolic element, in contrast to a strictly loxodromic one, can be presented as the composition of reflections in two hyperbolic planes. Those planes are either intersecting (the line of the intersection is called the axis of the elliptic element; this is the fixed point set under its action), or parallel (the common point of the planes, which lies on \overline{C} = ∂H^{3}, is the unique fixed point of the parabolic element), or disjoint, respectively (in the last case, the common perpendicular to the planes is the unique invariant line under the action of the hyperbolic element and is called the axis of this element).

An elliptic element is determined by its axis, the angle and the direction of rotation about this axis. An elliptic element of finite order is called primitive (or geometrically primitive), if it is conjugate, on \overline{C}, to a rotation of the form F (z) = exp(2πi/p)z, p ∈ N, and non-primitive otherwise. A non-primitive elliptic element of order p is conjugate to a rotation F (z) = exp(2πiq/p)z, where p, q ∈ N are relatively prime, q > 1 and q/p < 1/2.

Further information on Kleinian groups and hyperbolic geometry can be found in [2], [5], [6], [25], and [29].

We remind only the following Proposition 1.4. Let f and g be hyperbolic elements. Then tr[f, g] ∈ R if and only if the axes of f and g either lie in one hyperbolic plane or they are mutually orthogonal skew lines.

This is a special case of Theorem 1 in [20].

2. Main result

Suppose that both elements f and g of PSL(2, C) are hyperbolic and tr[f, g] = trfgf^{−1}g^{−1} is real.

By Proposition 1.4 the axes of f and g either lie in a hyperbolic plane (and then this plane is invariant under the action of Γ = ⟨f, g⟩; i.e., Γ is discrete if and only if it is Fuchsian), or those axes are mutually orthogonal skew lines. In the last case, necessary and sufficient conditions for Γ to be discrete are given by the following

Theorem 2.1. Let f, g ∈ PSL(2, C) be hyperbolic elements with mutually orthogonal skew axes. Then

(1) there exists a unique element h ∈ PSL(2, C) such that h^{2} = [f, g] and (hg)^{2} = 1; and

(2) Γ = ⟨f, g⟩ is discrete if and only if h is hyperbolic, parabolic, or a primitive elliptic element.

Proof. To prove the theorem we will use the methods expounded in papers [17]–[19].

1. The first step is to construct a group of reflections Γ^{∗} so that Γ is its subgroup of finite index. In this case, the question about discreteness of Γ is equivalent to the same question for Γ^{∗}. Simultaneously we prove conclusion (1) of the theorem.

We denote an element and its axis by the same letter. So, f and g are mutually orthogonal skew lines. Let e be the half-turn about the common perpendicular to f and g. There exist two other half-turns e_{f} and e_{g} such that

f = e_{f}e, g = e_{g}e.

We have

[f, g] = (e_{f}e)(e_{g}e)(ee_{f})(ee_{g}) = (fe_{g})^{2}.  (1)

It is clear that the axes e_{f} and e lie in one hyperbolic plane, are disjoint, and f is their common perpendicular. An analogous statement is true for the three axes e_{g}, e, and g.

Let λ, μ, σ, τ be hyperbolic planes defined as follows: λ contains e_{f} and is orthogonal to f; μ contains e_{g} and is orthogonal to g; σ contains e_{f}, e, and f; and finally τ contains e_{g}, e, and g. Note that the pairs of planes λ and σ, σ and τ, τ and μ intersect at an angle of π/2, λ and τ are disjoint (f is their common perpendicular), μ and σ are also disjoint (with g as their perpendicular). The planes λ and μ are either disjoint, or parallel, or intersecting.

Figure 1:

Let P_{0} denote the convex polyhedron (of infinite volume) bounded by λ, μ, σ and τ with edges e_{f}, e, e_{g}, and possibly λ ∩ μ, if this intersection is not empty (see Figure 1^{2}). If we denote by R_{λ}, R_{μ}, R_{σ}, R_{τ} the reflections in λ, μ, σ, τ respectively, then f = R_{λ}R_{τ}, e_{g} = R_{τ}R_{μ} and from (1) it follows that

[f, g] = (R_{λ}R_{μ})^{2}.

Denote by h the composition of reflections R_{μ} and R_{λ},

h = R_{λ}R_{μ}.

Then h is the very square root of [f, g] we need. Indeed, the product hg = (R_{λ}R_{μ})(R_{μ}R_{σ}) = R_{λ}R_{σ} = e_{f} is a half-turn. (Note that the other orientation preserving square root h′ of non-parabolic commutator [f, g] does not satisfy the condition that h′g is a half-turn. Namely, h′g is hyperbolic in case that [f, g] is elliptic, and h′g is π-loxodromic when [f, g] is hyperbolic.) We have proved conclusion (1) of the theorem.

Define

Γ̃ = ⟨f, g, e⟩ and Γ^{∗} = ⟨f, g, e, R_{τ}⟩ = ⟨R_{λ}, R_{μ}, R_{σ}, R_{τ}⟩.

Since efe^{−1} = f^{−1} and ege^{−1} = g^{−1}, we have Γ̃ = Γ ∪ Γe. It is clear that either Γ̃ = Γ (if e ∈ Γ), or Γ is an index 2 subgroup of Γ̃. Moreover, Γ̃ is the orientation preserving subgroup of Γ^{∗} of index 2. Thus Γ, Γ̃ and Γ^{∗} are either all discrete or all non-discrete.

2. Assume that h = R_{λ}R_{μ} is hyperbolic, parabolic, or a primitive elliptic element of order p (that means that λ and μ are disjoint, parallel, or intersecting at an angle of π/p, respectively). Then P_{0}, together with the reflections R_{λ}, R_{μ}, R_{σ}, R_{τ} in its faces, satisfies the hypotheses of Poincaré’s polyhedron theorem, and so Γ^{∗} = ⟨R_{λ}, R_{μ}, R_{σ}, R_{τ}⟩ is discrete. We have proved the “if” part of (2).

3. Assume that h = R_{λ}R_{μ} is neither hyperbolic, nor parabolic, nor a primitive elliptic element. We have two possibilities: either h is an elliptic element of infinite order; or h is a non-primitive elliptic element of finite order. In the first case, both Γ^{∗} and Γ are not discrete. In the second case, we need further investigation. To complete the proof of the theorem it suffices to prove the following

2All figures are done irrespective of any model of Lobachevsky space.

Lemma 2.2. Let f, g ∈ PSL(2, C) be hyperbolic elements with mutually orthogonal skew axes, and let h as above be a non-primitive elliptic element of finite order. Then Γ is not discrete.

Proof. We will prove it by contradiction. Assume that Γ is discrete, and h is a non-primitive elliptic element, that is, the planes λ and μ intersect at an angle of q(π/p), where p is the order of h, (p, q) = 1, 1 < q < p/2. Then Γ^{∗} (which is also discrete) contains reflections in q − 1 planes which pass through h and decompose the corresponding dihedral angle of P_{0} into q smaller angles of π/p.

Now we need an additional construction.

The lines e_{f} and h lying in λ are disjoint since they lie in disjoint planes σ and μ. Denote by ε the plane which passes through the common perpendicular to e_{f} and h and is orthogonal to λ. Then ε is orthogonal to e_{f} and h (Proposition 1.1), and consequently, to both μ and σ (Proposition 1.2). Therefore ε passes through the common perpendicular to μ and σ, i.e., through g (Proposition 1.3). Analogous argument shows that there exists a plane η which is orthogonal to λ, μ, and τ and passes through f (Figure 2). Denote by P_{c} the compact polyhedron bounded by λ, μ, σ, τ, ε, and η.

Figure 2:

Recall that Γ^{∗} contains the reflections R_{λ}, R_{μ}, R_{σ}, and R_{τ} (and does not contain the reflections in ε and η); moreover, Γ^{∗} contains reflections in q − 1 planes (q − 1 ≥ 1) passing through h and intersecting the interior of P_{c}. We call these q − 1 planes the additional reflection planes (through h).

Consider the planes ε and η. Set Γ_{ε}^{∗} = ⟨R_{λ}, R_{μ}, R_{σ}⟩, Γ_{η}^{∗} = ⟨R_{λ}, R_{μ}, R_{τ}⟩. It is easy to see that Γ_{ε}^{∗} keeps ε invariant, and Γ_{η}^{∗} keeps η invariant (since λ, μ, and σ are orthogonal to ε; and λ, μ, and τ are orthogonal to η). The additional reflection planes through h are orthogonal to both ε and η; and the reflections in them belong to both Γ_{ε}^{∗} and Γ_{η}^{∗}.

Figure 3:

Both Γ_{ε}^{∗} and Γ_{η}^{∗} are subgroups of Γ^{∗} (which is discrete by assumption), hence they are also discrete.

For convenience denote λ_{ε} = λ ∩ ε, μ_{ε} = μ ∩ ε, σ_{ε} = σ ∩ ε, λ_{η} = λ ∩ η, μ_{η} = μ ∩ η, τ_{η} = τ ∩ η (see Figure 3, where η and ε are shown).

We concentrate on ε, keeping in mind that analogous conclusions are valid also for η. Let ζ be the nearest to λ additional reflection plane through h. Then, ζ and σ are either disjoint, parallel, or intersecting. Denote by △_{ε} the triangle (compact or non-compact of finite or even infinite area) formed by λ_{ε}, σ_{ε}, and ζ_{ε} = ζ ∩ ε. Using discreteness of Γ_{ε}^{∗} and Knapp’s list [24], we have only those possibilities for △_{ε} that are presented in Figure 4 (in case when △_{ε} is compact).

Figure 4:

A priori, for every triangle △_{ε} with fixed angles there are as many different decompositions of ε (by the reflection planes that belong to Γ_{ε}^{∗}) as there are different q’s that are relatively prime to p and satisfy the inequality 1 < q < p/2. (We remind that qπ/p is the angle between λ and μ.)

We show below that in fact there is at most one decomposition for each admissible triangle △_{ε}.

First of all, for each △_{ε} we select exactly one decomposition D of ε that satisfies the following condition:

(C1 ) Among all decompositions with the same △_{ε}, D has the minimal number of additional reflection planes through h.

All possible types of decompositions satisfying (C1 ) are represented in Figure 5, where we suppose that case (i) contains subcases of non-compact triangle △_{ε} and of disjoint planes ζ and σ. Notice also that the minimal number of additional reflection planes through h in case (ii) equals two, because otherwise μ_{ε} intersects σ_{ε}.

We will denote the edges (and their lengths) and faces of the compact polyhedron P_{c} by \overline{g} = g ∩ P_{c}, \overline{λ}_{ε} = λ_{ε} ∩ P_{c}, \overline{τ}_{ε} = τ_{ε} ∩ P_{c}, …. It is easy to see that for every decomposition shown in Figure 5 we have:

(C2 ) \overline{σ}_{ε} meets at most one additional reflection plane.

Thus, (C2 ) follows from (C1 ). It is not difficult to see that (C2 ) holds even if (C1 ) does not occur. Indeed, one can check it by adding (to the corresponding decomposition in Figure 5) one or several lines passing through the vertex h ∩ ε.

Figure 5:

The similar condition is always realized for η:

(C2′) \overline{τ}_{η} meets at most one additional reflection plane.

So, we have proved that the discreteness of Γ^{∗} implies (C2 ) and (C2′). Finally, we show that decompositions of ε (by the reflections of Γ_{ε}^{∗}) other than in Figure 5 do not exist. Otherwise, at least two additional reflection planes intersect \overline{g} and, consequently, also intersect \overline{τ}_{η}, which contradicts condition (C2′).

It is clear that Figure 5 presents not only all decompositions of ε, but also those of η (by the reflection planes of Γ_{η}^{∗}).

Figure 6:

Since the reflection planes passing through h are the same for ε and η, a priori there are the following three possibilities for the pair {ε, η}:

⟨1⟩ {(i), (iii)} or {(iii), (i)};

⟨2⟩ {(ii), (ii)};
⟨3⟩ {(iii), (iii)}.

Note that (iv) is impossible (it could be combined only with (iii), but there p = 7).

To complete the proof of Lemma 2.2 it remains to show that the situations ⟨1⟩ – ⟨3⟩ cannot occur.

In ⟨3⟩, the additional reflection plane intersects e at a point, say x (Figure 6). We have two axes of elliptic elements of order 7 passing through x, what is impossible in a discrete group. (It is an easy corollary of the following fact. If a subgroup of a discrete isometry group of H^{3} keeps some sphere invariant, then this subgroup is a discrete group of isometries of 2-dimensional sphere. For our subgroup generated by the two elliptic elements of order 7 any sphere centered at x is invariant.)

In hyperbolic geometry a triangle is determined by its angles, therefore, in case ⟨2⟩, \overline{λ}_{ε} = \overline{μ}_{η}. Hence \overline{μ}_{ε} = \overline{λ}_{η} (see Figure 7). However, on the one hand, \overline{λ}_{ε} > \overline{μ}_{ε} (it is easy to see from Figure 5(ii) for the plane ε). Moreover, λ_{ε} is the common perpendicular to h and e_{f}, and, consequently, \overline{λ}_{ε} < \overline{λ}_{η}. We have a contradiction.

In case ⟨1⟩ we can restrict ourselves to handling the pair {(iii), (i)}. Denote by ζ_{σ} the line of intersection of the additional reflection plane with σ. First of all, ζ_{σ} cannot intersect e (otherwise, there are two intersecting axes, ζ_{σ} of order 7 and e of order 2, in the discrete group what is possible only if the angle of their

Figure 7:

intersection is equal to π/2, but then in σ we have a triangle with two right angles). The lines ζ_{σ} and e cannot be parallel, because none of the discrete groups in Euclidean plane contains elliptic element of order 7. The last possibility is that ζ_{σ} and e are disjoint. In this case, in the plane orthogonal to both ζ_{σ} and e we have a triangle with angles of π/2, π/7, φ. This triangle is an orthogonal section of an infinite triangle prism formed by reflection planes for which ζ_{σ} and e are lateral edges. Thus, this triangle have to be smaller than any other section of the prism. On the other hand, from discreteness of Γ^{∗}, we conclude that φ = π/n (n ≥ 3) or φ = 2π/7. However, all such triangles are larger than the one hatched on Figure 8 (the hatched triangle is a part of a triangle with angles π/2, π/7, π/3) and, hence, cannot be the orthogonal section of the prism. We arrived at a contradiction. The proof of Lemma 2.2 (and Theorem 2.1) is complete.

Figure 8:

Remark 2.3. It is easy to reformulate the above result in terms of parameters (β, β′, γ), see [20], Table 2.

Remark 2.4. Surprisingly, there are no discrete groups with two hyperbolic generators for which the above mentioned square root h of their commutator is a non-primitive elliptic element (compare this, for example, with the case of two elliptic generators).

3. Two-generator hyperbolic orbifolds of infinite volume

In this section we construct hyperbolic orbifolds that correspond to all the discrete groups described in Theorem 2.1.

Note that if h is a primitive elliptic element of order p, where p is odd, then there is an integer k such that [f, g]^{k} = h. Hence the elements h, e_{f} = hg, and e = e_{f}f belong to Γ. We have Γ̃ = ⟨f, g, e⟩ = Γ. To construct a fundamental polyhedron for Γ̃ = Γ we double P_{0} along its face σ and denote by P_{1} the doubled polyhedron. It is easy to see that P_{1} and the elements e, e_{f}, and g satisfy the hypotheses of Poincaré’s Polyhedron Theorem [5]. We obtain that Γ has the following presentation:

Γ = ⟨e, e_{f}, g | e^{2} = e_{f}^{2} = (e_{f}g)^{p} = (ge)^{2} = 1⟩.  (a)

To obtain the orbifold with boundary (H^{3} ∪Ω(Γ))/Γ (whose interior H^{3}/Γ is a hyperbolic orbifold and Ω(Γ) is the discontinuity set of Γ), we identify the equivalent faces of P_{1}. Note that all our orbifolds are embedded into S^{3}, and we draw them so that ∞ is a regular inner point of the orbifold (see Figure 9 (a)).

In the remaining cases, when h is hyperbolic, parabolic or a primitive elliptic element of even order p, for the construction of the fundamental polyhedron for Γ we double P_{1} (the fundamental polyhedron for Γ̃) along τ to obtain the polyhedron P_{2} which together with f and g satisfies the hypotheses of Poincaré’s Theorem.

Figure 9:

We see that Γ ≠ Γ̃ and (as a Kleinian group) has one of the following presentations:

Γ = ⟨f, g | (fgf^{−1}g^{−1})^{p/2} = 1⟩,  (b)

if [f, g] = h^{2} is elliptic;

Γ = ⟨f, g | (fgf^{−1}g^{−1})^{∞} = 1⟩,  (c)

if [f, g] = h^{2} is parabolic; and

Γ = ⟨f, g | ⟩,  (d)

if [f, g] = h^{2} is hyperbolic. For the corresponding orbifolds see Figure 9 (b, c, d, respectively).

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Elena Klimenko Department of Mathematics Technion–Israel Institute of Technology Haifa 32000 Israel klimenko@tx.technion.ac.il

Received July 10, 2000 and in final form February 9, 2001