Arithmeticity, Discreteness and Volume

F. W. Gehring  C. Maclachlan  G. J. Martin

A. W. Reid ^{∗}

April 27, 1995

Abstract

We give an arithmetic criterion which is sufficient to imply the discreteness of various two-generator subgroups of P SL(2, C). We then examine certain two-generator groups which arise as extremals in various geometric problems in the theory of Kleinian groups, in particular those encountered in efforts to determine the smallest co-volume, the Margulis constant and the minimal distance between elliptic axes. We establish the discreteness and arithmeticity of a number of these extremal groups, the associated minimal volume arithmetic group in the commensurability class and we study whether or not the axis of a generator is simple.

1 Introduction

In the papers [11], [13], [14], [15], [17] concerning geometric constraints on Kleinian groups, we identify many two-generator subgroups of P SL(2, C) which are candidates for various extremal problems. These include a family of extreme groups in which the minimum distance between non–intersecting elliptic axes is realized, the minimum co-volume Kleinian group, the Kleinian group in which the Margulis constant is realized, and a family of discrete groups in which the distance between the fixed points of tetrahedral, octahedral and icosahedral subgroups is also minimized; see also [9]. Not surprisingly all of these problems are closely related.

∗Research supported in part by grants from the U. S. National Science Foundation, the N.Z. Foundation of Research, Science and Technology, the Australian Research Council, the U. K. Royal Society and the U.K. Scientific and Engineering Research Council. We wish also to thank the University of Texas at Austin, the University of Auckland and the MSRI at Berkeley (NSF Grant DMS-90222140) for their support during part of this work.

We are grateful to D. J. Lewis who gave us a proof for Theorem 5.12 and to K. N. Jones who computed the co-volumes that appear in Table 11.

A principal obstruction to solving some of these questions was in establishing the discreteness of proposed candidates for the extremals. Often these groups were either two-generator groups or contained two-generator subgroups of finite index. In several cases we established the discreteness of these two-generator groups by an ad hoc method; see [13]. A main result of this paper gives a unified method, derived from elementary algebraic number theory, which deals with most of these extremal situations. As a consequence we find that many of the extremals are subgroups of arithmetic Kleinian or Fuchsian groups. These results are derived from a classical result in number theory whose application to proving groups discrete in P SL(2, C) appears to be new. When one considers how difficult it is in general to prove that a two-generator subgroup of P SL(2, C) is discrete, the criterion considered here seems surprisingly straightforward and elegant.

Most of the extremal examples mentioned above arise from a disk covering procedure based on complex iteration and a special semigroup of polynomials. This method is ideally suited for the discreteness criteria which we shall establish here. Hence we first outline this procedure in §2 in order to motivate the criteria which we then give and discuss at the end of §2; §3, §4 and §5 contain background material on arithmetic groups and the proofs for the results given in §2.

In §6 we apply these criteria to a family of two-generator groups ⟨f, g⟩ where f and g are elliptics of orders n and 2, n = 3, 4, 5, 6, 7. These groups resulted from the disk covering method discussed above while studying the possible distances between non-intersecting elliptic axes in a discrete group, possible minimum co-volume Kleinian groups, the Margulis constant and the length structure of closed geodesics.

The groups ⟨f, g⟩ in §6 are subgroups of arithmetic groups. In §7 and §8 we first compare the minimal co-volume of the arithmetic groups in which the groups ⟨f, g⟩ embed with the distance between the axes of the generators f and g. Using a computer program developed by K. Jones and the last author it is possible to determine which of the groups ⟨f, g⟩ have finite co-volume and hence are themselves arithmetic.

Finally motivated by the small volume problem for orbifolds, we derive in §9 and then apply in §10 arithmetic and geometric criteria to determine which of the groups ⟨f, g⟩ of §6 have f as a simple elliptic generator.

2 Disk covering method and discreteness

Let G = ⟨f, g⟩ be a two-generator subgroup of P SL(2, C). We associate to each such group three complex numbers called the parameters of G,

par(G) = (γ(f, g), β(f), β(g)) ∈ C^{3},  (2.1)

where

β(f) = tr^{2}(f) − 4, β(g) = tr^{2}(g) − 4, γ(f, g) = tr([f, g]) − 2

and [f, g] = fgf^{−1}g^{−1} is the multiplicative commutator. These three complex numbers determine G uniquely up to conjugacy whenever γ(f, g) ≠ 0; recall γ(f, g) = 0 implies the existence of a common fixed point. Conversely, every such triple determines a two-generator subgroup of P SL(2, C).

When γ ≠ 0, β, there is a natural projection from C^{3} to C^{2}, given by

(γ, β, β′) → (γ, β, −4),

which preserves discreteness; see [13]. That is, if (γ, β, β′) are the parameters for a discrete two-generator group, then so are (γ, β, −4). Hence in these circumstances one can always replace one of the generators in a discrete two-generator group by an element of order two without altering the commutator or discreteness. No such result holds for groups with more than two-generators [5].

We shall study the pairs of parameters (γ(f, g), β(f)) in C^{2} which can correspond to discrete groups ⟨f, g⟩. For the purposes of this discussion we specialize to the case where f is a primitive elliptic element of order n ≥ 3. Then

β = β(f) = −4 sin^{2}(π/n)

and for each n ≥ 3, we want to describe the set E_{n} of values in the complex plane which may be assumed by the commutator parameter γ(f, g) in the case when < f, g > is discrete. This set cannot have 0 as a limit point [13].

Next if

h = g ∘ f^{m_{1}} ∘ g^{−1} ∘ f^{m_{2}} ∘ g … f^{m_{n}} ∘ g^{ϵ}  (2.2)

where ϵ = (−1)^{n}, then it follows from [13] that

γ(f, h) = p(γ(f, g), β(f))  (2.3)

where p is a polynomial in the two variables with integer coefficients. Equation (2.3) shows that for such words h, the value of γ(f, h) depends only on γ(f, g) and β(f) and not on β(g). In particular, the family of such polynomials p is closed under composition in the first variable and so forms a polynomial semigroup P. Thus we may study the iterates of a specific polynomial and, in this way, generate a sequence of commutator traces which cannot accumulate at 0.

For example if h = gfg^{−1}fg and γ = γ(f, g), then

p(γ, β) = γ(1 + β − γ)^{2}.

Hence if f is elliptic of order 6, then β = −1 and p(γ, β) = γ^{3}. In this case if we set γ_{j} = γ(f, g_{j}) where

g_{j+1} = g_{j}fg_{j}^{−1}fg_{j} and g_{1} = g,

then γ_{j+1} = γ^{3^{j}} → 0 as j → ∞ if 0 < |γ| < 1, a contradiction. Thus γ = 0 or |γ| ≥ 1 and we obtain an analogue for elliptics of order 6 of the classical inequality of Shimizu–Leutbecher which yields the same conclusion whenever f is parabolic [26]. In particular,

E_{6} ⊂ C \ D_{6}

where D_{6} = {z : 0 < |z| < |}.

For each value of n ≥ 3 we can argue as above using iteration with an appropriate polynomial in P to obtain an open punctured disk D_{n} about 0 which does not contain the parameter γ(f, g). When n ≥ 7, the existence of such a neighborhood follows from Jørgensen’s inequality. Next if p(γ) = p(γ, β) is the polynomial in P which corresponds to the element h in (2.2), then the subgroup ⟨f, h⟩ cannot be discrete if

p(γ(f, g)) = p(γ(f, g), β(f)) = γ(f, h) ∈ D_{n}.

Hence p^{−1}(D_{n}) is an excluded region for γ(f, g) and we conclude that

E_{n} ⊂ C \ {∪p^{−1}(D_{n}) : p ∈ P}.

In this way we can cover regions of the complement of the set E_{n} of admissible values for the commutator parameter of groups with a generator of order n.

The program for describing these regions was begun in [13] and was significantly extended in [16]. Obtaining various geometric constraints for such groups amounts to showing that a certain region Ω lies in the complement of E_{n} except possibly for some exceptional values.

The result of this disk covering argument is illustrated below for the case where n = 3. All exceptional values within the region covered by the disks are illustrated and every value turns out to correspond to a subgroup of an arithmetic group. Many are in fact two-generator arithmetic groups themselves. That is, they are additionally of finite co-volume.

Possible values for commutator parameter when n=3

The closure of the hyperbolic line joining the two fixed points of a nonparabolic element f is called the axis of f, denoted by axis(f). The following formula yields collaring theorems from which volume estimates may be obtained; see [14] and [18].

Lemma 2.4 If f and g are non-parabolic, then the hyperbolic distance δ(f, g) between the axes of f and g is given by

cosh(2δ(f, g)) = |\frac{4γ(f, g)}{β(f)β(g)} + 1| + |\frac{4γ(f, g)}{β(f)β(g)}| .  (2.5)

The part of E_{n} in Ω is contained in the union X_{n} of the preimages of the punctures of D_{n} taken over all the polynomials p in P. Moreover E_{n} is a proper subset of X_{n} since the preimage of a puncture under one polynomial p in P can often be mapped into D_{n} by some other polynomial q in P, and hence will not be a point in E_{n}. At the end of such a covering argument one is left with a few points of X_{n} inside the given region Ω which may correspond to discrete groups. One must then analyze these exceptions and prove that either they are not discrete or, if they are discrete, use knowledge of these exceptional groups to compute the invariants being studied.

It is often not easy to decide whether a given point γ in X_{n} belongs to E_{n}, i.e. whether γ corresponds to discrete group with a generator of order n. However in our situation there exists a certain polynomial with coefficients in Z[β] for which γ is a root. Thus it is natural to look for arithmetic conditions which guarantee discreteness. The fortunate fact is that this works well for the small extremals, the general philosophy being that “small” implies “highly symmetric” which in turn implies “arithmetic”.

Under the assumptions of discreteness and finite co-volume on subgroups of P SL(2, R) and P SL(2, C), the results of [33] and [24] give conditions on the traces of elements in the subgroup which characterize, among Fuchsian and Kleinian groups, those which are arithmetic; see §4. Dropping the assumption of finite co-volume characterizes those which are subgroups of arithmetic Fuchsian or Kleinian groups. More importantly, the proofs in [33] and [24] make no use of the discreteness assumption; cf. [1]. Thus the arithmetic conditions on the traces can be used to deduce discreteness.

In this paper refinements of these arithmetic conditions which will guarantee discreteness are obtained. These are particularly applicable to twogenerator groups such as those corresponding to the points γ in X_{n} described above.

For example, suppose that G = ⟨f, g⟩ where f is a primitive elliptic element of order n ≥ 3 and g is elliptic of order 2. Then

β = β(f) = −4 sin^{2}(π/n)

is an algebraic integer in the field Q(cos(2π/n)), which is totally real with ϕ(n)/2 places. Let R_{n} denote the ring of integers in Q(cos(2π/n)) and note that R_{n} = Z[β]. The Galois conjugates of β, σ(β) = −4 sin^{2}(mπ/n) where (m, n) = 1, lie in the interval (−4, 0) with σ(β) ≤ β. Recall that G is determined up to conjugacy by β and γ(f, g). With this we prove the following criteria for discreteness. Recall that, for a number field K of degree n over Q, the n Galois embeddings σ : K → C give rise to valuations on K which fall into equivalence classes - the places - modulo the action of complex conjugation.

Theorem 2.6 Let G = ⟨f, g⟩ be a subgroup of P SL(2, C) with f a primitive elliptic element of order n ≥ 3, g an elliptic of order 2 and γ(f, g) ≠ 0, β(f). Then G is a subgroup of a discrete arithmetic group if

1. Q(γ(f, g), β(f)) has at most one complex place;

2. γ(f, g) is a root of a monic polynomial p(z) ∈ Z[β][z];

3. if γ(f, g) and γ̄(f, g) are not real, then all other roots of p(z) are real and lie in the interval (β, 0);

4. if γ(f, g) is real, then all other roots of p(z) are real and lie in the interval (β, 0).

The requirement that the field Q(γ(f, g), β(f)) has at most one complex place can, for β(f) as in Theorem 2.6, also be described in terms of the polynomial p(z). If in addition n = 3, 4 or 6, then β ∈ Z and the criteria admit a greatly simplified description. See Theorem 5.14 below.

3 A Criterion for Discreteness

In this section we prove a result that guarantees discreteness under certain conditions on the invariant trace field. Our result is related to a classical result in number theory and does not seem to be well known in the area of Kleinian groups. Indeed it is a generalization of the prototype result that yields discreteness for arithmetic subgroups of P SL(2, C). As we point out in [23] this theorem has a proof in the language of arithmetic groups which makes application more routine for our purposes; see also the discussion in §4. However we give here an elementary proof which illuminates the connection with the following well known facts from number theory.

Recall that if p is a monic irreducible polynomial over Z of degree n with roots α_{1}, …, α_{n}, then

p(z) = ∏_{i=1}^{n}(z − α_{i}) = z^{n} − s_{1}z^{n−1} + … (−1)^{k}s_{k}z^{n−k} + … (−1)^{n}s_{n} (3.1)

where s_{i} is the i^{th} symmetric polynomial in α_{1}, …, α_{n}. As a consequence we deduce the following easy lemma.

Lemma 3.2 There are only finitely many algebraic integers z of bounded degree such that z and all Galois conjugates of z are bounded.

Proof. Let z have degree n over Q. If z and its Galois conjugates are bounded, then the coefficients of the irreducible polynomial p of z are symmetric polynomials in the roots of p and hence are bounded. Therefore only finitely many integers can arise as coefficients of such a polynomial. □

Next we recall some notation from [28]. Let G be a finitely generated subgroup of P SL(2, C). The trace field of G is the field generated over Q by the set

tr(G) = {±tr(g) : g ∈ G}.

Since G is finitely generated, the subgroup G^{(2)} = ⟨g^{2} : g ∈ G⟩ is a normal subgroup of finite index with quotient group a finite abelian 2-group. Following [28] we call

kG = Q(tr(G^{(2)}))

the invariant trace-field of G. For any finite index subgroup G_{1} of a nonelementary group G one can show that Q(tr(G^{(2)})) ⊂ Q(tr(G_{1})); in [31] it is shown that kG is an invariant of the commensurability class.

Throughout the paper, we will use

c : C → C  (3.3)

to denote the complex conjugation map. With this notation we can establish the following criterion for discreteness.

Theorem 3.4 Let G be a finitely generated subgroup of P SL(2, C) such that

1. G^{(2)} contains elements g_{1} and g_{2} which have no common fixed point;

2. tr(G) consists of algebraic integers;

3. for each embedding σ : kG → C such that σ ≠ id or c, the set {σ(tr(f)) : f ∈ G^{(2)}} is bounded.

Then G is discrete.

Proof. Note that since G is finitely generated, so is G^{(2)} and so all traces in G^{(2)} are obtained from integral polynomials in a finite number of traces. Thus kG is a finite extension of Q.

It suffices to prove that the finite index subgroup G^{(2)} is discrete. Suppose that this is not the case and let f_{n} be a sequence of distinct elements converging to the identity in G^{(2)}. If z_{n} = tr(f_{n}) and z_{n,i} = tr([f_{n}, g_{i}]), then

β(f_{n}) = z_{n}^{2} − 4 → 0 and γ(f_{n}, g_{i}) = z_{n,i} − 2 → 0

for i = 1, 2 as n → ∞. Hence we may assume that |z_{n}| < K for some fixed constant K. Next by condition 3., |σ(z_{n})| < K_{σ} for each embedding σ ≠ id or c of kG, where K_{σ} is a constant which depends only on σ.

Let R = max{K, K_{σ}} where σ ranges over all embeddings σ ≠ id or c of kG. Then the algebraic integers z_{n} are of bounded degree and they and all their Galois conjugates are bounded in absolute value by R. By Lemma 3.2, the z_{n} assume only finitely many values. Thus for large n, β(f_{n}) = 0 and f_{n} is parabolic with a single fixed point w_{n}.

Next we can apply the above argument to the algebraic integers z_{n,i} to conclude that γ(f_{n}, g_{i}) = 0 for i = 1, 2 and large n. This then implies that g_{1} and g_{2} each have w_{n} as a common fixed point for large n contradicting condition 1. □

4 Arithmetic Kleinian Groups

In this section we first recall some terminology about quaternion algebras and then discuss Theorem 3.4 in the language of these algebras and arithmetic groups. In particular we give a version of Theorem 3.4 which is readily applicable, especially in the context of two-generator groups; see §5.

We begin with some facts about quaternion algebras; see [34] for details. Let k be a number field, let ν be a place of k, i.e. an equivalence class of valuations on k and denote by k_{ν} the completion of k at ν. If B is a quaternion algebra over k, we say that B is ramified at ν if B ⊗_{k} k_{ν} is a division algebra of quaternions. Otherwise ν is unramified.

In case ν is a place associated to a real embedding of k, B is ramified if and only if B ⊗_{k} k_{ν} ≅ H, where H is the Hamiltonian division algebra of quaternions.

It is straightforward to check whether a quaternion algebra is ramified at a real place. Recall, following [34], the Hilbert symbol (\frac{a,b}{k}) corresponding to the quaternion algebra {1, i, j, ij} defined over the field k with i^{2} = a, j^{2} = b and ij = −ji.

Lemma 4.1 Let B = (\frac{a,b}{k}). Then B is ramified at a real place ν corresponding to the real embedding σ if and only if σ(a) and σ(b) are negative.

Proof. By definition, B is ramified at ν whenever B ⊗_{k} k_{ν} ≅ H. This tensor product is isomorphic to

(\frac{σ(a), σ(b)}{σ(k)}) ⊗_{σ(k)} R

or simply (\frac{σ(a),σ(b)}{R}). This, in turn, is isomorphic to H exactly when σ(a) and σ(b) are negative since one can remove squares without affecting the isomorphism class of the quaternion algebra; cf. [34]. □

We now give the definition of an arithmetic Kleinian group. Let k be a number field with one complex place and A a quaternion algebra over k ramified at all real places. Next let ρ be an embedding of A into M(2, C), O an order of A and O^{1} the elements of norm 1 in O. Then ρ(O^{1}) is a discrete subgroup of SL(2, C) and its projection to P SL(2, C), P ρ(O^{1}), is an arithmetic Kleinian group. The commensurability classes of arithmetic Kleinian groups are obtained by considering all such P ρ(O^{1}).

Arithmetic Fuchsian groups arise in a similar manner. In this case the number field is totally real and the algebra ramified at all real places except the identity.

In [24] (resp. [33]), it is shown that two arithmetic Kleinian groups (resp. arithmetic Fuchsian groups) are commensurable up to conjugacy if and only if their invariant quaternion algebras are isomorphic; see also [3].

To state the characterization theorems in [24] and [33] referred to earlier, we first define, following Bass [1],

AG = {∑ a_{i}g_{i} : a_{i} ∈ Q(tr(G)), g_{i} ∈ G}

for any finitely generated non-elementary subgroup of SL(2, C). Then AG is a quaternion algebra over Q(tr(G)). Recall that kG = Q(tr(G^{(2)})) and AG^{(2)} is then the invariant quaternion algebra since, for a finitely generated nonelementary subgroup G of P SL(2, C), the pair (AG^{(2)}, kG) is an invariant of the commensurability class of G. See [28]. Additionally, if tr(G) consists of algebraic integers, then as in [1]

OG = {∑ a_{i}g_{i} | a_{i} ∈ R_{Q(tr(G))}, g_{i} ∈ G}

is an order in AG, where R_{Q(tr(G))} denotes the ring of integers in Q(tr(G)).

Theorem 4.2 Let G be a Kleinian (resp. Fuchsian) group of finite covolume. Then G is arithmetic if and only if the following conditions are satisfied:

1. kG is an algebraic number field;

2. tr(G) consists of algebraic integers;

3. for every Q-isomorphism σ : kG → C such that σ ≠ id, c, σ(tr(G^{(2)})) is bounded in C.

Sketch of Proof. It is a simple matter to show that arithmetic Kleinian and Fuchsian groups have these properties. Note that if γ ∈ H^{1} ≅ S^{3} then tr(γ) ∈ [−2, 2].

Since G has finite co-volume, it is finitely generated and non-elementary. Only these two properties are used in the proof, except at the final stage. Condition 1. actually follows from 2. and the fact that G is finitely generated. We thus obtain a quaternion algebra AG^{(2)} over kG with AG^{(2)} ⊂ M(2, C). Furthermore, each embedding σ : kG → C extends to an embedding of AG^{(2)} in M(2, C).

If γ ∈ SL(2, C) has eigenvalues u, u^{−1} whose absolute values are not one, then it follows that {tr(γ^{m}) : m ∈ Z} is unbounded since

|tr(γ^{m})| ≥ ||u|^{m} − |u|^{−m}|.

Thus in the case at hand condition 3. implies that kG has exactly one complex place if kG is not real and is totally real if kG is real. Also with γ as above, tr(γ) ∈ [−2, 2] and condition 3. also implies that AG^{(2)} is ramified at all real places of kG ≠ id. Finally condition 2. shows that OG^{(2)} is an order of AG^{(2)} and, of course, G^{(2)} ⊂ (OG^{(2)})^{1}, which is a discrete arithmetic Kleinian or Fuchsian group. Since G has finite co-volume, it will be commensurable with (OG^{(2)})^{1} and so be arithmetic. □

In practice, direct application of Theorem 4.2 (or Theorem 3.4) is hard, the problem being to establish the boundedness of the traces at real embeddings. However, as a corollary of the proof of Theorem 4.2, we obtain the following more useful method for proving groups discrete.

Theorem 4.3 Let G be a finitely generated non-elementary subgroup of P SL(2, C) such that

1. kG has exactly one complex place or is totally real;

2. tr(G) consists of algebraic integers;

3. AG^{(2)} is ramified at all places of kG, ≠ {id, c}.

Then G is a subgroup of an arithmetic Kleinian or Fuchsian group.

If g_{1}, g_{2} ∈ G^{(2)} have no common fixed point and g_{1} is not parabolic, the proof of Theorem 4.2 shows that AG^{(2)} is spanned by 1, g_{1}, g_{2}, g_{1}g_{2} over kG; see [20], [24] and [33]. A basis in standard form can then be obtained yielding the Hilbert symbol

(\frac{β(g_{1}), γ(g_{1}, g_{2})}{kG}) .

Now if f, g ∈ G are a pair of non-commuting elements with f not parabolic and f, g not of order 2, then

β(f^{2}) = (β(f) + 4)β(f), γ(f^{2}, g^{2}) = (β(f) + 4)(β(g) + 4)γ(f, g)

so that

AG^{(2)} ≅ (\frac{(β(f) + 4)β(f), (β(f) + 4)(β(g) + 4)γ(f, g)}{kG}) .  (4.4)

See [20]. If we choose, as we may do, f and g to have equal traces, then by removing squares in the Hilbert symbol we obtain

AG^{(2)} ≅ (\frac{(β(f) + 4)β(f), γ(f, g)}{kG}) .  (4.5)

Finally note that if kG = Q(tr(G)) we can simplify still further to

AG^{(2)} ≅ (\frac{β(f), γ(f, g)}{kG}) .  (4.6)

With these representations of AG^{(2)}, condition 3. of Theorem 4.3 can be readily checked using Lemma 4.1. Thus in summary we have the following result.

Theorem 4.7 Let G be a non-elementary finitely generated subgroup of P SL(2, C) with a pair of non-commuting elements f and g, where f is not parabolic and f and g are both not of order 2. Then G is discrete if

1. Q(tr(G)) is a finite extension of Q;

2. tr(G) consists of algebraic integers;

3. kG has one complex place or is totally real;

4. with AG^{(2)} ≅ (\frac{a, b}{kG}) described at (4.4), (4.5), (4.6), then σ(a) and σ(b) are negative for each embedding σ of kG ≠ id, c.

5 Two-generator groups

Here we specialize the discreteness theorems in the previous sections to the case where G is a two-generator group and, in particular, where both generators are elliptic. In these cases, both the invariant field and the invariant quaternion algebra will be readily described in terms of the parameters of the group.

As stated above, kG = Q(tr(G^{(2)})) is an invariant of the commensurability class. In fact, it is shown in [31] that the field kG coincides with the field

Q({tr^{2}(g) : g ∈ G}) = Q({β(g) : g ∈ G}).

Actually, it is not difficult to see that if G is generated by elements g_{1}, g_{2}, …, g_{n} with β(g_{i}) ≠ −4 for i = 1, 2, …, n, then

kG = Q(tr(G^{sq}))  (5.1)

where G^{sq} = ⟨g_{1}^{2}, g_{2}^{2}, …, g_{n}^{2}⟩; see [20]. For two-generator groups this has the following consequence.

Lemma 5.2 Let G = ⟨f, g⟩ be a subgroup of SL(2, C) with β(f) ≠ −4 and β(g) ≠ −4. Then

kG = Q(β(f), β(g), β(fg^{−1}) − γ(f, g)).  (5.3)

Proof. The trace of any element in a two-generator group ⟨ϕ, ψ⟩ group is given by a polynomial with integer coefficients in tr(ϕ), tr(ψ), tr(ϕψ^{−1}). See, for example [21]. Thus Q(tr(G^{sq})) is Q(tr(f^{2}), tr(g^{2}), tr(f^{2}g^{−2})). Then since

tr(f^{2}) = β(f) + 2, tr(g^{2}) = β(g) + 2,

the desired conclusion follows from the identity

tr(f^{2}g^{−2}) = β(fg^{−1}) + 2 − γ(f, g). □

The following corollary will be of use to us.

Corollary 5.4 Let G = ⟨f, g⟩ be a group with β(f) ≠ −4 and γ(f, g) ≠ 0, β(f). If G_{1} = ⟨f, gfg^{−1}⟩, then

kG_{1} = Q(β(f), γ(f, g)).  (5.5)

Proof. Applying Lemma 5.2 in the case where the generators are f and gfg^{−1} and f as above, we see that

kG_{1} = Q(β(f), β(fgf^{−1}g^{−1}) − γ(f, gfg^{−1})).  (5.6)

An easy calculation shows

β(fgf^{−1}g^{−1}) − γ(f, gfg^{−1}) = γ(f, g)(β(f) + 4).

Next the hypotheses on β(f) and γ(f, g) imply that f and gfg^{−1} do not commute. Thus kG_{1} = Q(γ(f, g), β(f)). □

Corollary 5.7 Let G = ⟨f, g⟩ be a group with β(f) ≠ −4, β(g) = −4 and γ(f, g) ≠ 0, β(f). Then

kG = Q(γ(f, g), β(f)).

Proof. Since g has order two, G_{1} = ⟨f, gfg^{−1}⟩ has index two in ⟨f, g⟩. Thus kG_{1} = kG being an invariant of the commensurability class and the result follows from Corollary 5.4. □

The hypotheses of Corollary 5.7 imply that the groups G = ⟨f, g⟩ and G_{1} = ⟨f, gfg^{−1}⟩ are simultaneously discrete or nondiscrete. For two-generator groups G we have also identified kG in terms of the parameters of G. We have therefore proved the following result.

Corollary 5.8 Let G = ⟨f, g⟩ be a group with par(G) = (γ, β, −4) where β ≠ −4, 0 and γ ≠ 0, β. Then kG = Q(γ, β) and

AG^{(2)} = (\frac{β(β + 4), γ(γ − β)}{Q(γ, β)}).  (5.9)

Proof. Again G_{1} = ⟨f, gfg^{−1}⟩ is a subgroup of index two in G with

par(G_{1}) = (γ(γ − β), β, β).

Our hypotheses imply this group is nonelementary and kG_{1} = kG = Q(γ, β). The Hilbert symbol follows from (4.5). □

From Theorem 4.7, we can now deduce as a corollary the following sufficient condition for a two-generator group to be a subgroup of an arithmetic Kleinian group, and hence discrete.

Theorem 5.10 Let G = ⟨f, g⟩ be a subgroup of P SL(2, C) with par(G) = (γ, β, −4) where β ≠ 0 and γ ≠ 0, β. Then G is a discrete group if

1. γ and β are algebraic integers;

2. the field kG = Q(γ, β) has exactly one complex place or is totally real;

3. if kG has a complex place, then −4 < σ(β) < 0 and σ(γ(γ − β)) < 0 for all real embeddings σ;

4. if kG is totally real, then −4 < σ(β) < 0 and σ(γ(γ − β)) < 0 for all real embeddings σ ≠ id.

Indeed G is a subgroup of an arithmetic Fuchsian or Kleinian group with invariant trace field Q(γ, β) where the Hilbert symbol of the invariant quaternion algebra is

(\frac{(β + 4)β, γ(γ − β)}{Q(γ, β)}).

Proof. The traces of f, fg satisfy monic quadratic polynomials whose coefficients lie in Z[γ, β]. Since the traces of elements in G are polynomials with integer coefficients in tr(f), tr(g), tr(fg), the first hypothesis implies that all traces of elements of G are algebraic integers. Therefore the elements of G^{(2)} also have algebraic integer traces. By Corollary 5.8, the Hilbert symbol for AG^{(2)} is as claimed, and so by the last pair of assumptions we see that AG^{(2)} is ramified at all real places, except possibly in the case when kG is totally real.

By our previous discussion it follows that OG^{(2)} is an order of AG^{(2)}, and therefore OG^{(2)}^{1} yields an arithmetic Kleinian or Fuchsian group via the construction discussed previously. Thus G^{(2)} is discrete and hence so is G.

In the proof we constructed the arithmetic group OG^{(2)}^{1} which contains G^{(2)}. To get an arithmetic group containing G consider the group

Norm(OG^{(2)}) = {x ∈ AG^{(2)} : xOG^{(2)}x^{−1} = OG^{(2)}}.

The image of this group is an arithmetic subgroup of PSL(2, C) (see [3] for example). As G^{(2)} is normal in G, it follows that G is a subgroup of an arithmetic Kleinian group. □

In the cases where f is primitive elliptic of order n and g is elliptic of order 2, then β = −4 sin^{2} π/n is a totally real algebraic integer so γ must satisfy a monic polynomial in Z[β][z] to be an algebraic integer. For all embeddings σ of Q(γ, β), −4 < σ(β) ≤ β < 0 and so Theorem 2.6 is an immediate corollary of Theorem 5.10.

We now refine these results to obtain more easily applicable criteria for discreteness. We first recall some basic results concerning field extensions.

Suppose that K is a finite extension of Q, that γ is algebraic over K and that p(z) is the minimum polynomial of γ over K with deg(p) = m > 1. Let σ : K → C be an embedding. Then there are exactly m embeddings τ : K(γ) → C such that τ|K = σ and these are uniquely determined by τ(γ) = γ′ where γ′ runs through the roots of σ(p(z)). See for example [10].

Lemma 5.11 If K(γ) has exactly one complex place, then K must be totally real, i.e. every embedding of K into C must be real.

Proof. If σ : K → C is a complex embedding, so is σ̄ and σ̄ ≠ σ. Then by the above remarks, there exist 2m embeddings of K which are complex. □

Theorem 5.12 Let γ and β be algebraic integers such that γ ∉ Q(β)⊂Q(γ, β) and let p(z, β) be the minimum polynomial of γ over Q(β). Then Q(γ, β) has one complex place if and only if

1. β is totally real, i.e. all of the Galois conjugates of β, β=β_{1}, β_{2}, …, β_{n} are real;

2. the polynomial q(z) = p(z, β_{1})p(z, β_{2}) ⋯ p(z, β_{n}) has exactly one pair of complex conjugate roots and all other roots real.

Proof. For j = 1, 2, …, n let σ_{j} denote the embedding of Q(β) into C defined by σ_{j}(β) = β_{j} and suppose that τ is an embedding of Q(γ, β) into C. Then τ is determined by the images τ(γ), τ(β). Now τ(β) = σ_{j}(β) for some j in which case τ(γ) is a root of σ_{j}(p(z, β)) = p(z, β_{j}). Thus conditions 1. and 2. imply that there are precisely two complex embeddings Q(γ, β) into C.

Conversely if Q(γ, β) has exactly one pair of complex embeddings into C, then by Lemma 5.11 each embedding of Q(β) into C must be real and so an embedding τ of Q(γ, β) into C will be complex precisely when a root of σ_{j}(p(z, β)) = p(z, β_{j}) is complex. □

Note that with γ, β as in Theorem 5.12, Q(γ) must have at least one complex place and so by Lemma 5.11, Q(γ) = Q(γ, β).

Now specialize again to the cases where G = ⟨f, g⟩ with f a primitive elliptic element of order n ≥ 3, g elliptic of order 2 and γ ≠ 0, β. Thus Q(β) is totally real and the Galois conjugates β_{k} = −4 sin^{2}(kπ/n) where (k, n) = 1 and 1 ≤ k ≤ n/2 so that β_{k} = β(f^{k}). Note that −4 < β_{k} < β < 0 for k ≠ 1. Thus from Theorem 5.10 and 5.12, we obtain the following result.

Theorem 5.13 Let G = ⟨f, g⟩ be a subgroup of P SL(2, C) with f a primitive elliptic of order n ≥ 3, g elliptic of order 2 and γ(f, g) ≠ 0, β(f). Then G is a discrete subgroup of an arithmetic group if

1. γ(f, g) is the root of a polynomial p(z, β(f)) in z and β(f) with integer coefficients;

2. the roots of p(z, β(f^{k})) are real and lie in the interval (β(f^{k}), 0) for all k such that (k, n) = 1 and 2 ≤ k ≤ n/2;

3. if γ(f, g) and γ̄(f, g) are complex, then all other roots of p(z, β(f)) lie in the interval (β(f), 0);

4. if γ(f, g) is real, then all other roots of p(z, β(f)) lie in the interval (β(f), 0).

Finally if n = 3, 4 or 6, then β ∈ Z and we have the following result.

Theorem 5.14 Let G = ⟨f, g⟩ be a subgroup of P SL(2, C) with f a primitive elliptic of order n = 3, 4 or 6, g an elliptic of order 2 and γ(f, g) ≠ 0, β(f). Then G is a discrete subgroup of an arithmetic group if

1. γ(f, g) is the root of a polynomial p(z) with integer coefficients;

2. if γ(f, g) and γ̄(f, g) are complex, then all other roots of p(z) lie in the interval (β(f), 0);

3. if γ(f, g) is real, then all other roots of p(z) lie in the interval (β(f), 0).

6 Examples

In this section we use the discreteness criteria in §5 to identify a number of interesting two-generator subgroups of arithmetic groups. Some of these were shown in [13], [14] and [15] to be extremal for geometric properties such as minimum axial distance, minimum co-volume and the Margulis constant. A number of these groups are of finite co-volume themselves and are therefore arithmetic; recall that our methods identify the invariant trace field and quaternion algebra.

The groups we consider are generated by a pair of elliptic elements f and g where f is of order n ≥ 3 and g of order 2. We present these in tabular form for the cases n = 3, 4, 5, 6, 7. By virtue of Theorems 5.13 and 5.14 all of these are subgroups of arithmetic groups, and hence discrete. We discuss in §7 and §8 their relations with small volume hyperbolic 3-orbifolds and in §10 we determine the groups in which the generator f is a simple elliptic.

We begin with a pair of preliminary remarks. Suppose that (γ, β, −4) are the parameters of a discrete two–generator group G = ⟨f, g⟩ with f of order n. Then the Lie product G^{∗} of G [22] and the conjugations Ḡ and Ḡ^{∗} of G and G^{∗} by reflection in the real axis are discrete two-generator groups with

par(G^{∗}) = (β − γ, β, −4), par(Ḡ) = (γ̄, β, −4), par(Ḡ^{∗}) = (β − γ̄, β, −4).

See, for example, [13]. Moreover, if G is a subgroup of an arithmetic group, then the same is true of G^{∗}, Ḡ and Ḡ^{∗}. Hence it is sufficient to consider only those values of γ for which

−2 sin^{2}(π/n) = β/2 ≤ Re(γ), 0 ≤ Im(γ).

Next if γ ≠ 0, β, then by Lemma 2.4 the hyperbolic distance δ = δ(f, g) between the axes of the generators f and g of G is given by

cosh(2δ) = \frac{|γ − β| + |γ|}{|β|}.  (6.1)

In particular, δ = 0 if and only if ⟨f, g⟩ is elementary. Hence in our case, δ is a measure of how much ⟨f, g⟩ differs from an elementary group. We see from (6.1) that the corresponding axial distances for G^{∗}, Ḡ and Ḡ^{∗} are equal to that for G.

We now give for n = 3, 4, 5, 6, 7 tables of groups G_{n,i} generated by elliptics of orders n and 2 together with 1. the approximate values of their commutator parameters γ_{n,i},

2. the minimum polynomial p_{n,i} for γ_{n,i} over Q(β),

3. the approximate distance δ_{n,i} between the axes of the generators of G_{n,i}.

When n = 3, 4, 6, p_{n,i} coincides with the minimum polynomial q_{n,i} for γ_{n,i} over the field Q considered in the next section.

These groups were identified by the disk covering procedure described in §2. The resulting diagrams, showing the only possible values for the parameter γ_{n,i} for the cases where n = 3, 4, 5, 6, are given at the end of this paper in §11. For an account of the calculations on which these diagrams are based, see [16].

We begin with tables for the groups G_{3,i} and G_{4,i}.

Table 1 - Groups G_{3,i}
i  γ_{3,i}  p_{3,i}  δ_{3,i}
1  −1  z + 1  0
2  −.3819  z^{2} + 3z + 1  0
3 −1.5 + .6066i  z^{4} + 6z^{3} + 12z^{2} + 9z + 1  .1970
4 −.2118 + .4013i  z^{4} + 5z^{3} + 7z^{2} + 3z + 1  .2108
5 −.5803 + .6062i  z^{3} + 4z^{2} + 4z + 2  .2337
6 −1.1225 + .7448i  z^{3} + 5z^{2} + 8z + 5  .2448
7 −.3376 + .5622i  z^{3} + 3z^{2} + 2z + 1  .2480
8 −.9236 + .8147i  z^{4} + 5z^{3} + 8z^{2} + 6z + 1  .2740
9 −1.5 + .8660i  z^{2} + 3z + 3  .2746
10 −.7672 + .7925i  z^{3} + 4z^{2} + 5z + 3  .2770
11 .0611 + .3882i  z^{4} + 5z^{3} + 6z^{2} + 1  .2788
12  .2469  z^{3} + 4z^{2} + 3z − 1  .2831
13 −.5284 + .7812i z^{6} + 8z^{5} + 24z^{4} + 35z^{3} + 28z^{2} + 12z + 1 .2944
14 −.1153 + .5897i  z^{3} + 3z^{2} + z + 1  .2970

Table 2 - Groups G_{4,i}
i  γ_{4,i}  p_{4,i}  δ_{4,i}
1  −1  z + 1  0
2  −.5 + .8660i  z^{2} + z + 1  .4157
3 −.1225 + .7448i  z^{3} + 2z^{2} + z + 1  .4269
4  −1 + i  z^{2} + 2z + 2  .4406
5 −.6588 + 1.1615i  z^{3} + 3z^{2} + 4z + 3  .5049
6 .2327 + .7925i  z^{3} + z^{2} + 1  .5225
7 −.2281 + 1.1151i  z^{3} + 2z^{2} + 2z + 2  .5297
8 .4196 + .6062i  z^{3} + z^{2} − z + 1  .5297
9  i  z^{2} + 1  .5306
10  .6180  z^{2} + z − 1  .5306
11 −1 + 1.2720i z^{4} + 4z^{3} + 7z^{2} + 6z + 1 .5306
12 −.4063 + 1.1961i z^{4} + 3z^{3} + 4z^{2} + 4z + 1 .5345
13 .7881 + .4013i  z^{4} + z^{3} − 2z^{2} + 1  .6130

Tables 3 and 5 contain the groups for n = 5 and 7. Here p_{n,i} is a polynomial in z and β to which Theorem 5.13 applies.

Table 3 - Groups G_{5,i}
i  γ_{5,i}  p_{5,i}  δ_{5,i}
1  −.3819  z − β − 1  0
2 −.6909 + .7228i  z^{2} − βz + 1  .4568
3 .1180 + .6066i  z(z − β − 1)^{2} − β − 1  .5306
4 −.1909 + .9815i  z^{2} − (β + 1)z + 1  .6097
5  .6180  z − β − 2  .6268
6 .2527 + .8507i  z(z − β − 1)^{2} + 1  .6514
7 −.6909 + 1.2339i  z^{2} − βz + 2  .6717
8 −.3819 + 1.2720i z^{3} − (2β + 1)z^{2} + (β^{2} + β + 2)z − 2β − 1 .6949
9 .1180 + 1.1696i z^{3} − (2β + 2)z^{2} + (β^{2} + 2β + 2)z − β .7195
10 −.0817 + 1.2880i z^{4} − (2β + 1)z^{3} + (β^{2} + β + 2)z^{2} − 2βz + 1 .7273
11 .6180 + .7861i  z^{3} − (2β + 3)z^{2} + (β^{2} + 3β + 2)z + 1  .7323
12 .8776 + .5825i  z(z − β)(z − β − 2)^{2} + 1  .7725

Table 4 - Groups G_{6,i}
i  γ_{6,i}  p_{6,i}  δ_{6,i}
1 −.5 + .8660i  z^{2} + z + 1  .6584
2  i  z^{2} + 1  .7642
3  .5 + .8660i  z^{2} − z + 1  .8314
4 −.5 + 1.3228i  z^{2} + z + 2  .8500
5 −.2150 + 1.3071i z^{3} + z^{2} + 2z + 1 .8539
6 .3411 + 1.1615i  z^{3} + z + 1  .8786
7  1  z − 1  .8813
8 .8774 + .7448i  z^{3} − z^{2} + 1  .9106
Table 5 - Groups G_{7,i}
i  γ_{7,i}  p_{7,i}  δ_{7,i}
1  .2469  z − β − 1 .5452
2 −.3765 + .9264i z^{2} − βz + 1 .8162
3  1.2469  z − β − 2 1.0704

We conclude this section with two examples to illustrate how the above tables are constructed. The groups we consider, G_{3,2} and G_{5,2}, are those where the minimum distances between the axes of elliptics of orders 3 and 2 and orders 5 and 2 are realized.

In the first example, β = β(f) = −3 and the disk covering procedure shows that the polynomial

z(z + 3)(z^{2} + 3z + 3)(z^{4} + 6z^{3} + 12z^{2} + 9z + 1)^{2}(z^{4} + 6z^{3} + 12z^{2} + 9z + 3),

corresponding to the word hfh^{−1}fhfh^{−1} where h = gfg^{−1}fgf^{−1}g^{−1}, maps

γ_{3,2} = −3/2 + i\sqrt{(2\sqrt{5} − 3)/4} ≈ −1.5 + .606658i

onto 0. Then

p_{3,2}(z) = z^{4} + 6z^{3} + 12z^{2} + 9z + 1

is the minimum polynomial for γ_{3,2} and it is easy to check that p_{3,2} has real roots −.13324, −2.86676 which lie in the interval (−3, 0). Hence we can apply Theorem 5.14 to conclude that G_{3,2} is a subgroup of an arithmetic group and hence discrete.

In the second case,

β = β(f) = (\sqrt{5} − 5)/2, β(f^{2}) = (−\sqrt{5} − 5)/2

and from the disk covering argument we find that

p_{5,2}(z, β) = z^{2} − βz − β

is the minimum polynomial for

γ_{5,2} = (\sqrt{5} − 5)/4 + i\sqrt{(5\sqrt{5} − 7)/8} ≈ −.690983 + .722871i

over Q(β). Then γ_{5,2} and γ̄_{5,2} are the roots of p_{5,2}(z, β(f)), the roots −.301522 and −3.31651 of p_{5,2}(z, β(f^{2})) are both real and the desired conclusion follows from Theorem 5.13.

7 Small volume arithmetic orbifolds

We now discuss the minimal co-volume arithmetic Kleinian groups that the groups G_{n,i} embed in. We deal with the case n = 3 in some detail and in §8 merely state the salient information for the others in tabulated form.

We begin with some comments on the more detailed arithmetic structure associated to these groups. For this we need some additional results and terminology, for which we refer the reader to [34].

Let B be a quaternion algebra over a number field k. The isomorphism class of B is determined by those places of k, both Archimedean and non-Archimedean, at which B is ramified. The number of ramified places is always even. The non-Archimedean places correspond to the prime ideals in R, the ring of integers in k. If O is an order in B, then its discriminant d(O) is an ideal of R. Furthermore, O will be a maximal ideal precisely when d(O) is the product of those ideals at which B is ramified. If M is an order of B such that M ⊂ O, then d(O) | d(M). In the cases where O is a free R-module with basis {f_{1}, f_{2}, f_{3}, f_{4}}, the discriminant can be determined from the formula

d(O)^{2} = ⟨Det(tr_{B}(f_{i}f_{j}))⟩

where tr_{B} : B → k is the reduced trace.

The following lemma will prove useful in our calculations.

Lemma 7.1 Let G be a finite co-volume Kleinian group such that tr(G) consists of algebraic integers and let R denote the ring of integers in Q(tr(G)). If ⟨f, g⟩ is a nonelementary subgroup of G, then R[1, f, g, fg] is an order of AG.

Proof. Recall that 1, f, g, fg span AG over the trace field. Thus since R[1, f, g, fg] contains a basis of AG, is finitely generated and contains R, it suffices to prove that all products of the basis elements can be expressed as R-combinations of the basis elements. There are several obvious ones and these together with the following identities prove the lemma.

f^{2} = tr(f)f − 1,
f^{2}g = tr(f)fg − g,
(fg)^{2} = tr(fg)fg − 1,
fgf = −tr(g)1 + tr(fg)f + g,
gf + fg = (tr(fg) − tr(f)tr(g))1 + tr(g)f + tr(f)g. □

We adopt some notation that will be used in the next two sections. For n = 3,4,5,6,7 let kG_{n,i} and A_{n,i} denote the invariant trace field and invariant quaternion algebra of G_{n,i}.

We now fix attention on the case of n = 3 and, in what follows, suppress the subscript 3 for convenience of notation. Next, referring to Table 1 in the cases i=1, 2 and 12, since the corresponding commutator parameter is real, one checks easily that these groups are the (2,3,4) spherical triangle group, the (2,3,5) spherical triangle group and the (2,3,7) Fuchsian triangle group. Also in the case i=13, the invariant trace field has degree 6 and we will not make any further comment on this case. Thus we concentrate on the cases when the commutator parameter is not real and the relevant polynomials have degree 2, 3 and 4.

Let O_{i} denote the R_{kG_{i}}-submodule of A_{i} generated over R_{kG_{i}} by the elements

{1, g_{i}f_{i}g_{i}^{−1}, f_{i}^{−1}, g_{i}f_{i}g_{i}^{−1}f_{i}^{−1}};

note that since f_{i} has order 3, f_{i} and g_{i}f_{i}g_{i}^{−1} lie in G_{i}^{(2)}. Lemma 7.1 shows that O_{i} is an order of A_{i}.

By Theorem 5.10, the algebras A_{i} have the following Hilbert symbols,

(\frac{−3, (γ_{i} + 3)γ_{i}}{kG_{i}} ).

It is an easy calculation using the comments above to compute the discriminant of O_{i}, namely

d(O_{i}) = ⟨tr([f_{i}^{−1}, g_{i}f_{i}g_{i}^{−1}]) − 2⟩ = ⟨γ_{i}(γ_{i} + 3)⟩.

We first deal with the case of the quartics and cubics.

Lemma 7.2 For each i ≠ 10, corresponding to a cubic or quartic, O_{i} is a maximal order.

Proof. Using the earlier remarks we need to compute the ramification of these quaternion algebras and compare them with γ_{i}(γ_{i} + 3). Since each ramified prime must divide d(O_{i}), it is expeditious to calculate d(O_{i}) in the first instance.

In all the quartic cases, a simple calculation shows that γ_{i}(γ_{i} + 3) is a unit, so that d(O_{i}) = R_{kG_{i}}, from which it follows that A_{i} cannot have any finite ramification and O_{i} is maximal.

A variation of this argument works in the cubic cases, i ≠ 10. For the norm of γ_{i}(γ_{i} + 3) in each of these cases is a rational prime so that d(O_{i}) is a prime ideal P_{p_{i}}. As the cardinality of the set of places ramified in any A_{i} is even and as each A_{i} here is ramified at exactly one real place, A_{i} must be ramified at at least one prime ideal. Thus it is ramified at precisely the one ideal P_{p_{i}} and O_{i} is maximal. For later computations, we note that p_{5} = 2, p_{6} = 5, p_{7} = 5, p_{14} = 2. □

For the case i = 10 we argue as follows. The discriminant of kG_{10} is -31 and by the fact that there is a unique cubic with one complex place and discriminant -31, an alternative description of kG_{10} is kG_{10} = Q(u) where u^{3} + u + 1 = 0; cf [29]. The ring of integers in this field has a unique prime ideal P_{3} of norm 3 and P_{3} =< γ_{10} >. Now γ_{10} + 3 also has norm 3 so that γ_{10}(γ_{10} + 3) = vγ_{10}^{2} where v is a unit. Thus d(O) = P_{3}^{2}. As in the above proof A_{10} must be ramified at at least one prime and so must be ramified at precisely the one finite place corresponding to P_{3}. Note that O_{10} is not a maximal order in this case. □

We now compute maximal discrete groups in which the groups G_{i}, with i as indicated above, are subgroups. To do this we make use of the description of maximal groups due to Borel [3]. First we consider only cubics and quartics and we assume that i ≠ 10. The case i = 10 will be dealt with separately.

Borel’s classification of maximal groups in the commensurablity class says that if O is a maximal order of the quaternion algebra B, then the group P ρ(Norm(O)) (recall the proof of Theorem 5.10) is a maximal group of minimal co-volume in that commensurability class. We denote this maximal group by Γ_{∅,∅} in the notation of [3]. Borel actually shows there are infinitely many maximal groups in the commensurability class of an arithmetic group; see the case of i = 10 below.

As in the proof of Theorem 5.10, since g_{i} normalizes ⟨f_{i}, g_{i}f_{i}g_{i}^{−1}⟩, we deduce that G_{i} < P ρ(Norm(O_{i})). By Lemma 7.2, for i ≠ 10 O_{i} is maximal and we obtain the following result.

Corollary 7.3 For i ≠ 10, G_{i} is a subgroup of the group P ρ(Norm(O_{i})) = Γ_{∅,∅}^{(i)}.

For the case i = 10, we utilize more of Borel’s construction of maximal groups referred to above. We briefly recall the relevant points. We recommend the reader have [3] at hand. In what follows B is a quaternion algebra over the number field k with one complex place, which is ramified at all real places. Let V denote the set of all finite places of k, and Ram_{f}(B) denote the finite places which ramify B.

Borel proves that to any pair S and S′ of finite (possibly empty), disjoint subsets of V \ Ram_{f}(B), one can associate a group Γ_{S,S′}. Moreover it is shown in [3] any arithmetic Kleinian group Γ in the commensurability class determined by B is conjugate to a subgroup of some Γ_{S,S′}. We will not make use of this explicitly here.

Let ν ∈ V \ Ram_{f}(B) and denote by C_{ν} be the Bruhat-Tits Building of SL(2, k_{ν}), which in these cases is a tree. The vertices of this tree are maximal orders of M(2, k_{ν}), and the SL(2, k_{ν}) action on the vertices forms 2 orbits. The groups Γ_{S,S′} are described in terms of vertex and edge stabilizers of these trees. Since B is unramified at ν we get B_{ν} = B ⊗_{k} k_{ν} ≅ M(2, k_{ν}). Thus for any group in the commensurability class determined by B we have an induced action on C_{ν}.

In our case, since d(O_{10}) is divisible only by P_{3}, if ν ∈ V \ P_{3}, then O_{10}, localized at ν is maximal; see [34]. Given any element of Norm(O_{10}), the image of this in A_{10,ν} stabilizes the vertex of C_{ν} associated to O_{10} localized at ν. It now follows from this and Borel’s description of maximal groups that P ρ(Norm(O_{10})) ⊂ Γ_{∅,∅}^{10}; see section 4 of [3]. Hence we can conclude that Corollary 7.3 also holds for i = 10.

To compute the minimal co-volumes in the cubic and quartic cases we use the formula of [3].

Quartic cases

We have seen from the proof of Lemma 7.2 that in the case of i = 3, 4, 8 and 11 the quaternion algebras A_{i} are unramified at all finite places. It is an easy calculation to show that the class number of these quartic fields is 1 and further that the structure of the group of units of these fields implies that the co-volume of the groups Γ_{∅,∅}^{(i)} is

\frac{|d_{kG_{i}}|^{3/2}ζ_{kG_{i}}(2)}{2^{7}π^{6},}

where d_{kG_{i}} is the discriminant and ζ_{kG_{i}}(s) is the Dedekind zeta function of kG_{i}; cf. [3]. Reasonably good estimates for ζ_{kG_{i}}(2) are obtained by considering primes of small norm in kG_{i} in the first few terms of the Euler product expansion of ζ_{kG_{i}}(2).

1. The polynomial z^{4} +6z^{3} +12z^{2} +9z +1 has discriminant −275, and this is the smallest discriminant of a field with one complex place; cf. [19]. As shown in [7] for example this is the invariant quaternion algebra of the smallest volume orientable arithmetic hyperbolic 3-orbifold. This minimal volume is approximately 0.03905 . . ..

2. The polynomial z^{4} + 5z^{3} + 7z^{2} + 3z + 1 has discriminant −283 and hence the discriminant of kG_{4} coincides with this. There is a unique quartic field of this discriminant; cf. [19]. The minimal co-volume in the commensurability class determined by A_{4} is approximately 0.0408, the second smallest volume known. This orbifold is commensurable with (5, 1)-Dehn surgery on the figure eight knot complement [6].

3. The other two quartics have discriminants −491 and −563, respectively. By comparing with [19], these are the discriminants of the invariant trace-fields. The smallest co-volume in the commensurability classes are approximately 0.1028 … and 0.1274 …, respectively.

Cubic cases

The corresponding volume formula in this case is

\frac{|d_{kG_{i}}|^{3/2}ζ_{kG_{i}}(2)(NP − 1)}{2^{6}π^{4},}

using the same terminology as above. The additional term NP denotes the norm of the prime ideal P ramifying the algebra in each case.

1. The polynomial z^{3}+5z^{2}+8z+5 has discriminant −23 which is necessarily the discriminant of the invariant trace-field. The minimal co-volume of a group in this commensurability class is approximately 0.07859. This orbifold is commensurable with the Weeks’ manifold of volume 0.942707 …

There is a unique field of discriminant −23. Hence a non-real root of the polynomial z^{3} + 3z^{2} + 2z + 1 also generates kG_{6}. In fact, up to conjugation, G_{7} is a subgroup of the same maximal group as contains G_{6}.

2. The polynomial z^{3} +4z^{2} +5z +3 has discriminant −31 which also must coincide with the discriminant of kG_{10}. Following the proof of Lemma 7.2, we observed that A_{10} is ramified at the place ν_{3} corresponding to P_{3}. From our comments above G_{10} is a subgroup of the minimal co-volume group which in this case is approximately 0.06596.

(In unpublished calculations we have shown that this orbifold is commensurable with (8, 1)-Dehn surgery on the figure-eight knot complement, and contains the orbifold group of (3, 0),(3, 0)-Dehn surgery on the Whitehead link as a subgroup of index 8. See also [20])

3. The polynomial z^{3} +4z^{2} +4z +2 has discriminant −44, which coincides with the discriminant of kG_{5}. The smallest co-volume of a group in the commensurability class is approximately 0.066194.

4. The polynomial z^{3} + 3z^{2} + z + 1 has discriminant −76 which is the discriminant of kG_{14}. The smallest co-volume in this commensurability class is 0.1642.

This leaves the case of i = 9 for which the relevant polynomial is quadratic. A root of this polynomial generates the the field Q(\sqrt{−3}). Moreover notice that directly from the polynomial we can read off that z^{2} + 3z = −3. Hence the algebra is

(\frac{−3, −3}{Q(\sqrt{−3})}).

By [34], this is isomorphic to M(2, Q(\sqrt{−3})). Indeed in this case, the trace field of G_{9} is Q(\sqrt{−3}) and therefore one easily deduces from [28] for example that G_{9} is conjugate to a subgroup of P SL(2, O_{3}) and hence of the minimal co-volume group PGL(2, O_{3}). This group has co-volume 0.08457 … and is the smallest volume of an orientable cusped hyperbolic 3-orbifold [27].

8 Axial distance and small volumes

In this section, we summarize calculations similar to those carried out in §7 for the cases where n = 4, 5, 6, 7, omitting those where the degree of the minimum polynomial exceeds four. There are some additional complications which we comment on below, but we make no attempt at detailing all the calculations.

In each case, the Hilbert symbol of the invariant quaternion algebra is easily computed in terms of the parameter γ_{n,i} as

(\frac{−4 sin^{2}(2π/n), γ_{n,i}(γ_{n,i} + 4 sin^{2}(π/n))}{kG_{n,i}}).  (8.1)

As in the case n = 3, we construct from the two-generator group ⟨f, g⟩ with f^{n} = g^{2} = 1 a suitable order which enables us to determine the finite ramification of the quaternion algebra and a small co-volume group, containing the normalizer of the order, in which the group G_{n,i} lies.

Now for n = 3, 4, 5, 6, 7 we give tables for the groups G_{n,i} of §6 with

1. the minimum polynomial q_{n,i} over Q for the commutator parameter γ_{n,i} of G_{n,i},

2. the discriminant d_{G_{n,i}} of the trace-field kG_{n,i},

3. the finite places Ram_{f} where the invariant quaternion algebra ramifies, 4. the approximate distance δ_{n,i} between the axes of the generators of

G_{n,i},

5. the approximate smallest co-volume V_{n,i} of an arithmetic Kleinian group in which we know G_{n,i} can be embedded.

Table 6 - Co-volume of group containing G_{3,i}

i  q_{3,i}  d_{kG_{3,i}} Ram_{f} δ_{3,i} V_{3,i}
1  z + 1  −− −−  0  S_{4}
2  z^{2} + 3z + 1  −− −−  0  A_{5}
3  z^{4} + 6z^{3} + 12z^{2} + 9z + 1  −275 ∅ .1970 .0390
4  z^{4} + 5z^{3} + 7z^{2} + 3z + 1  −283 ∅ .2108 .0408
5  z^{3} + 4z^{2} + 4z + 2  −44 P_{2} .2337 .0661
6  z^{3} + 5z^{2} + 8z + 5  −23 P_{5} .2448 .0785
7  z^{3} + 3z^{2} + 2z + 1  −23 P_{5} .2480 .0785
8  z^{4} + 5z^{3} + 8z^{2} + 6z + 1  −563 ∅ .2740 .1274
9  z^{2} + 3z + 3  −3  ∅ .2746 .0845
10  z^{3} + 4z^{2} + 5z + 3  −31 P_{3} .2770 .0659
11  z^{4} + 5z^{3} + 6z^{2} + 1  −491 ∅ .2788 .1028
12  z^{3} + 4z^{2} + 3z − 1  −− −− .2831 Fuch.
13 z^{6} + 8z^{5} + 24z^{4} + 35z^{3} + 28z^{2} + 12z + 1 ?  ?  .2944 ?
14  z^{3} + 3z^{2} + z + 1  −76 P_{2} .2970 .1654

Table 7 - Co-volume of group containing G_{4,i}

i  q_{4,i}  d_{kG_{4,i}} Ram_{f}  δ_{4,i} V_{4,i}
1  z + 1  −−  −−  0  S_{4}
2  z^{2} + z + 1  −3 {P_{4}, P_{3}} .4157 .1268
3  z^{3} + 2z^{2} + z + 1  −23  P_{8}  .4269 .1374
4  z^{2} + 2z + 2  −4  ∅  .4406 .2289
5  z^{3} + 3z^{2} + 4z + 3  −31  P_{3}  .5049 .2968
6  z^{3} + z^{2} + 1  −31  P_{3}  .5225 .2968
7  z^{3} + 2z^{2} + 2z + 2  −44  P_{2}  .5297 .0661
8  z^{3} + z^{2} − z + 1  −44  P_{2}  .5297 .0661
9  z^{2} + 1  −4  ∅  .5306 .1526
10  z^{2} + z − 1  −−  −−  .5306 Fuch.
11 z^{4} + 4z^{3} + 7z^{2} + 6z + 1 −400  ∅  .5306 .0717
12 z^{4} + 3z^{3} + 4z^{2} + 4z + 1 −331  ∅  .5345 .4475
13  z^{4} + z^{3} − 2z^{2} + 1  −283  ∅  .6130 .3475

Table 8 - Co-volume of group containing G_{5,i}

i  q_{5,i}  d_{kG_{5,i}} Ram_{f}  δ_{5,i} V_{5,i}
1  z^{2} + 3z + 1  −−  −−  0  A_{5}
2  z^{4} + 5z^{3} + 7z^{2} + 5z + 1  −475  ∅  .4568 .0933
3  z^{4} + 4z^{3} + 2z^{2} + z + 1  −275  ∅  .5306 .0390
4  z^{4} + 3z^{3} + 3z^{2} + 3z + 1  −275  ∅  .6097 .0390
5  z^{2} + z − 1  −−  −−  .6268 Fuch.
6 z^{6} + 6z^{5} + 11z^{4} + 8z^{3} + 7z^{2} + 7z + 1  ?  ?  .6514 ?
7  z^{4} + 5z^{3} + 9z^{2} + 10z + 4  −775 {P_{4}, P′_{4}} .6717 .461
8  z^{4} + 6z^{3} + 12z^{2} + 14z + 11  −400  ∅  .6949 .0717
9  z^{4} + 4z^{3} + 4z^{2} + 5z + 5  −475  ∅  .7195 .0933
10 z^{6} + 6z^{5} + 12z^{4} + 16z^{3} + 17z^{2} + 8z + 1 ?  ?  .7273 ?
11  z^{4} + 2z^{3} − 2z^{2} + 2z + 1  −400  ∅  .7323 .0717
12 z^{6} + 5z^{5} + 3z^{4} − 8z^{3} + z^{2} + 8z + 1  ?  ?  .7725 ?

Table 9 - Co-volume of group containing G_{6,i}

i  q_{6,i}  d_{kG_{6,i}} Ram_{f}  δ_{6,i} V_{6,i}
1  z^{2} + z + 1  −3  ∅  .6584 .0845
2  z^{2} + 1  −4 {P_{9}, P_{2}} .7642 .3053
3  z^{2} − z + 1  −3  ∅  .8314 .1691
4  z^{2} + z + 2  −7 {P_{2}, P′_{2}} .8500 .5555
5 z^{3} + z^{2} + 2z + 1 −23  P_{27}  .8539 .5106
6  z^{3} + z + 1  −31  P_{3}  .8786 .3298
7  z − 1  −−  −−  .8813 Fuch.
8  z^{3} − z^{2} + 1  −23  P_{27}  .9106 .5106

Table 10 - Co-volume of group containing G_{7,i}

i  q_{7,i}  d_{kG_{7,i}} Ram_{f} δ_{7,i}  V_{7,i}
1  z^{3} + 4z^{2} + 3z − 1  −− −− .5452 Fuch.
2 z^{6} + 7z^{5} + 17z^{4} + 21z^{3} + 17z^{2} + 7z + 1 ?  ?  .8162  ?
3  z^{3} + z^{2} − 2z − 1  −− −− 1.0704 Fuch.

Notes

1. In these tables, we have adopted the notation of labeling a prime ideal of norm p^{n} by P_{p^{n}}. In the cases considered, this uniquely determines the ideal except for the indicated cases where P_{2}, P′_{2} or P_{4}, P′_{4} both appear.

2. We have not dealt with the cases where [Q(γ) : Q] > 4.

Remarks

1. As remarked in §7, any subgroup of an arithmetic Kleinian group is contained in a group Γ_{S,S′} [3]. In all cases when n = 3, applying this to G_{3,i} we have S = ϕ. That is not the situation in many cases when n = 4, 5, 6, but in most cases S consists of a single prime ideal. The volume of Γ_{S,S′} is readily calculated from the formulas given in [3].

2. A detailed knowledge of subgroups commensurable with tetrahedral groups enable some groups to be identified immediately by their commutator parameter and so the minimum volume of a group in which they lie is quickly determined. (See comments after the remarks on the tables). The arithmetic nature of tetrahedral groups is discussed in [25] and [30] from which much of the algebraic data of Table 8 is taken.

3. Table 7, n=4. The order we have considered is (suppressing sub-

scripts)

R_{kG}[1, f^{2}, (gf)^{2}, f^{2}(gf)^{2}]

which has discriminant ⟨2γ(γ + 2)⟩. Only in the cases i = 2, 3 is this order maximal. In these cases, the group G_{4,i} is contained in a Γ_{ϕ,ϕ} as is also the case for i = 7, 8 following an argument similar to that used in the case of G_{3,10} in §7. In the cases i = 4, 5, 6, 12, 13, S consists of a single prime, while in the case i = 9, S may consist of two primes. The groups G_{4,1} and G_{4,11} are subgroups of groups commensurable with tetrahedral groups.

4. Table 8, n = 5. Referring to Table 3, note that the polynomials p_{5,i} admit z + 1 as a factor in the cases i = 3,8,9,10,11,12. Using this, the field discriminants of Table 8 are readily determined. The order here is chosen to be

R_{kG}[1, gfg^{−1}, f^{−1}, [g, f]]

which has discriminant ⟨γ(γ − β)⟩ where β = (−5 + √5)/2. From this, as before, we determine the finite ramification of the algebra, which here, is only non-empty in the case i = 7. All other groups arise in considering cocompact tetrahedral groups.

5. Table 9, n = 6. Now we consider the order

R_{kG}[1, f^{2}, gf^{−2}g^{−1}, f^{2}gf^{−2}g^{−1}]

which has discriminant ⟨9γ(γ+1)⟩. For i = 2, 5, 8, the order is maximal so that S = ϕ, while when i = 3, 4, 6, S consists of at most one prime ideal. The group G_{6,1} is commensurable with a tetrahedral group. See also [12].

We now comment on the relationship between some of these groups and tetrahedral groups. Recall by a tetrahedral group we mean the orientationpreserving subgroup of index 2 in the group generated by reflections in the faces of tetrahedron in H^{3}, where some vertices may be ideal. Justifications for the comments below can be deduced for example from [30] where, as here, the notation for the tetrahedra is that of [4].

The groups PGL(2, Z[i]) and PGL(2, O_{3}) are tetrahedral, the tetrahedrons being T [3, 2, 2; 4, 2, 4] and T [3, 2, 2; 6, 2, 3]. In the tables these arise in the cases of G_{4,9} and PGL(2, O_{3}) occurs in the cases G_{3,9} and G_{6,1}. Other noncocompact groups which arise although not tetrahedral, contain a tetrahedral group of index 2. The case G_{4,4} yields a group containing the tetrahedral group T [4, 2, 2; 4, 2, 4] of index 2, and the case G_{6,3} yields a group containing the tetrahedral group T [2, 2, 3; 2, 6, 3] of index 2.

In the case of order 5 the algebras defined over the quartic fields listed in Table 8 except i = 7 yield groups commensurable with cocompact tetrahedral groups. In the case where the discriminant is −400 the tetrahedral group is T [2, 2, 4; 2, 3, 5] (which also arises in the case G_{4,11}), the discriminants −275 and −475 yield groups commensurable with the tetrahedral groups T [2, 2, 3; 2, 5, 3] and T [2, 2, 5; 2, 3, 5] respectively. The case T [2, 2, 3; 2, 5, 3] also arises in the case G_{3,3}.

Finally the order 6 case G_{6,4} yields a group commensurable with the tetrahedral group T [2, 3, 4; 2, 3, 4].

Recently K. N. Jones and the fourth author have developed a computer program to study explicitly how the geometry and topology of certain arithmetic Kleinian groups varies as the number theoretic data is varied. Among other things it constructs a fundamental polyhedron for the action of certain unit groups of orders and their normalizers in H^{3}, it determines the co-volume and it computes a presentation. We present in Table 11 the results produced by that program for the two-generator groups discussed in this paper. Since all but G_{4,11}, G_{5,7}, G_{6,4} and the Fuchsian examples turn out to have finite co-volume, we conclude that, with these exceptions, all these groups are themselves arithmetic.

Theorem 8.2 The following two-generator Kleinian groups G_{n,i} generated by elliptics of order n and 2 are arithmetic.

1. G_{3,i}, i = 3, …, 14, i ≠ 12, 13,

2. G_{4,i}, i = 2, …, 13, i ≠ 10, 11, 3. G_{5,i}, i = 2, …, 11, i ≠ 5, 6, 7, 10,

4. G_{6,i}, i = 1, …, 8, i ≠ 4.

Table 11 - Co-volumes of G_{n,i}

i G_{3,i} G_{4,i} G_{5,i}  G_{6,i}
1 S_{4} S_{4} A_{5} .2537
2 A_{4} .2537 .0933 .6106
3 .0390 .1374 .0390 .5074
4 .0408 .4579 .4686  ∞
5 .1323 .5936 Fuch. 1.0212
6 .1571 .5936  ?  1.3193
7 .1571 .7943 ∞  Fuch.
8 .1274 .2647 .8612 1.0212
9 .3383 .9159 1.1199 −−
10 .2638 Fuch. ? −−
11 .2056 ∞ .8612 −−
12 Fuch. .8951 ? −−
13 ? .3475 −−  −−
14 .3308 −− −−  −−

We conclude this section with some philosophical remarks. Several of the algebras and small volume groups arise in the considerations of [8]. This work identifies the smallest volume of an orientable arithmetic hyperbolic 3-manifold, namely that of the Weeks manifold. The content of this article together with [7] and the works [11]–[18] seem to indicate a convergence of arithmetic and geometric ideas. In particular the smallest volume orientable arithmetic orbifold arises as the quotient of hyperbolic 3-space H^{3} by a Z_{2}- extension of the tetrahedral group T [2, 3, 3; 2, 5, 3] (cf. [7]) and it would seem that this article and [11]—[16] are “converging on” the arithmetic orbifold just described as the overall smallest one, as is conjectured.

9 Criteria for simple axes

Suppose that f is an elliptic element of order n in a Kleinian group G. Then f is simple if axis(f) is precisely invariant, that is, if for each h ∈ G

h(axis(f)) = axis(hfh^{−1})

either coincides with or is disjoint from axis(f). If f is not simple, then one of the following is true [2] and [26].

1. The axes of f and hfh^{−1} intersect in H^{3} and G contains a subgroup isomorphic to the regular solid group A_{4}, S_{4} or A_{5}.

2. The axes of f and hfh^{−1} have one endpoint in common and G contains a subgroup isomorphic to the (3,3,3), (2,4,4) or (2,3,6) euclidean triangle group.

We show here that if G is a Kleinian group of finite co-volume, then each of these conditions forces the invariant quaternion algebra of G to satisfy certain criteria. Hence if the algebra fails to satisfy these criteria, then every elliptic in G of order n ≥ 3 must be simple.

Case where axes of f and hfh^{−1} intersect in H^{3}

Let H denote Hamilton’s quaternions so that H = (\frac{−1, −1}{R}) and let σ

denote the embedding σ : H^{1} → SL(2, C) given by

σ(a_{0} + a_{1}i + a_{2}j + a_{3}ij) = ( \begin{matrix}a_{0} + a_{1}i & a_{2} + a_{3}i \\ −a_{2} + a_{3}i & a_{0} − a_{1}i \end{matrix} ) ,

where H^{1} is the group of elements of norm 1. If n denotes the norm on H, then there is an epimorphism

Φ : H^{1} → SO(3, R)

where SO(3, R) is represented as the orthogonal group of the quadratic subspace V of H spanned by {i, j, ij}, that is, the pure quaternions, equipped with the restriction of the norm form, so that n(x_{1}i+x_{2}j+x_{3}ij) = x_{1}^{2}+x_{2}^{2}+x_{3}^{2}. The mapping Φ is defined by Φ(α) = ϕ_{α} where

ϕ_{α}(β) = αβα^{−1} α ∈ H^{1} β ∈ V.

The kernel of Φ is {±1}. The binary tetrahedral group is a central extension of an element of order 2 by the tetrahedral group and can be faithfully represented in H. This is also true for the binary octahedral group and the binary icosahedral group as will now be shown.

If the tetrahedron has its vertices at

i + j + ij, i − j − ij, −i + j − ij, −i − j + ij,

then ϕ_{α_{1}} is a rotation of order 2 about the axis through the edge mid-point i if α_{1} = i, while ϕ_{α_{2}} is a rotation of order 3 about the axis through the vertex i + j + ij when α_{2} = (1 + i + j + ij)/2. The binary tetrahedral group Γ_{1} is thus generated by α_{0} = −1, α_{1}, α_{2} in H^{1}.

If the cube has its vertices at

±i ± j ± ij,

then ϕ_{α_{3}} is a rotation of order 4 about the axis through the mid-point i of a face when α_{3} = (1+i)/\sqrt{2}. Thus the binary octahedral group Γ_{2} is generated by α_{0}, α_{2}, α_{3}.

If the regular icosahedron has its vertices at

±i ± j ± ij, ±τi ± τ^{−1}j, ±τj ± τ^{−1}ij, ±τij ± τ^{−1}i,

then ϕ_{α_{4}} is a rotation of order 5 about the axis through the mid-point of the face with vertices

i + i + ij, i + j − ij, τi + τ^{−1}j, τj + τ^{−1}ij, τj − τ^{−1}ij,

where α_{4} = (τ + τ^{−1}i + j)/2 and τ = (\sqrt{5} + 1)/2. The binary icosahedral group Γ_{3} is then generated by α_{0}, α_{2}, α_{4}.

The groups P σ(Γ_{1}) ≅ A_{4}, P σ(Γ_{2}) ≅ S_{4}, P σ(Γ_{3}) ≅ A_{5}, where P is the projection P : SL(2, C) → P SL(2, C), are said to be in standard form in P SL(2, C). If G is a discrete group which contains a finite subgroup F isomorphic to one of these regular solid groups, then G can be conjugated so that F is in standard form. Note that if G contains any of these finite groups, it will contain a subgroup isomorphic to A_{4}.

Lemma 9.1 Let G be a Kleinian group of finite co-volume with invariant quaternion algebra A and number field k. If G contains a subgroup isomorphic to A_{4}, then

A ≅ (\frac{−1, −1}{k}).  (9.2)

In particular, the only finite primes at which A can be ramified are the dyadic primes.

Proof. Suppose that G contains a subgroup isomorphic to A_{4}. Since A_{4} has no subgroup of index 2 and A_{5} is simple, we have A_{n} = A_{n}^{(2)} for n = 4 or 5. So if A_{n} ⊂ G, then A_{n} ⊂ G^{(2)}. Thus by conjugation, we can assume that σ(Γ_{1}) ⊂ G where P G = G^{(2)}. Now

A = {∑ a_{i}g_{i} : a_{i} ∈ k, g_{i} ∈ G}.

Let A_{0} = (\frac{−1, −1}{Q}). Then

A_{0} ≅ {∑ a_{i}g_{i} : a_{i} ∈ Q , g_{i} ∈ σ(Γ_{1})}

since 1, i, j, ij ∈ Γ_{1}. Now the quaternion algebra

{∑ a_{i}g_{i} : a_{i} ∈ k g_{i} ∈ σ(Γ_{1})}

lies in A, is isomorphic to A_{0} ⊗_{Q} k and is 4-dimensional. Thus

A ≅ (\frac{−1, −1}{k}).

Finally it follows from this form of the Hilbert symbol, that A splits over all P-adic fields k_{P}, with P non-dyadic, that is, primes not dividing 2. □

Lemma 9.3 Let G be as in Lemma 9.1. If G contains a subgroup isomorphic to A_{5} and if [k : Q] = 4, then A has no finite ramification.

Proof. A can at worst have dyadic finite ramification. Note that, since G must contain an element of order 5, Q(√5) ⊂ k. There is a unique prime P in Q(\sqrt{5}) such that P | 2. So if P ramifies or is inert in k | Q(\sqrt{5}), then there will only be one dyadic prime in k at which A cannot be ramified for parity reasons. Suppose then that P splits as P_{1}P_{2} so that k_{P_{1}} ≅ k_{P_{2}} ≅ Q(\sqrt{5})_{P}. But, again for parity reasons, the quaternion algebra

(\frac{−1, −1}{Q(\sqrt{5})})

splits in the field Q(\sqrt{5})_{P}. Hence

(\frac{−1, −1}{k})

splits in k_{P_{1}} and k_{P_{2}} and A has no finite ramification. □

Lemma 9.1 has the following partial converse.

Lemma 9.4 Suppose that k has exactly one complex place. If

A ≅ (\frac{−1, −1}{k}),

then there is an arithmetic Kleinian group in the commensurability class defined by A which contains S_{4}. Furthermore if Q(\sqrt{5}) ⊂ k, then there is a group in the commensurability class containing A_{5}.

Proof. To exhibit a group containing S_{4} we proceed as follows. Let R_{k} denote the ring of integers of k and let

O = R_{k}[1, i, j, (1 + i + j + ij)/2].

It is easily checked that O is an order in A. Note that Γ_{1} ⊂ O^{1}. Furthermore the element 1 + i ∈ O normalizes O^{1}. Thus if ρ is a representation of A into M(2, C), then

P ρ(O^{1}) ⊂ ⟨P ρ(O^{1}), P ρ(1 + i)⟩ = G

as a subgroup of index 2. But P ρ(1 + i) also normalizes P ρ(Γ_{1}) and

⟨P ρ(Γ_{1}), P ρ(1 + i)⟩ ≅ S_{4}

from the description above. Thus S_{4} ⊂ G.

Next, in the case where Q(√5) ⊂ k, if we let

O = R_{k}[1, i, 1/2(τ + τ^{−1}i + j), 1/2(−τ^{−1} + τi + ij)],

then O is an order in A. Note that the sum of the last two elements above is τ^{−1} + 1/2(1 + i + j + ij) so that Γ_{3} ⊂ O^{1}. Hence A_{5} ⊂ P ρ(O^{1}) as required. □

Case where axes of f and hfh^{−1} have one common endpoint

This case is handled by the following result.

Lemma 9.5 Let G be a subgroup of an arithmetic Kleinian group with invariant quaternion algebra A over k. If G contains a subgroup isomorphic to the (3,3,3), (2,4,4) or (2,3,6) euclidean triangle group, then

A ≅ M(2, Q(\sqrt{−1})) or A ≅ M(2, Q(\sqrt{−3})).  (9.6)

Proof. Since G contains a parabolic, a well known criterion for non-cocompact arithmetic groups implies that A ≅ M(2, (\sqrt{−d})) for some square-free integer d. Then it follows easily from [28] or [32] that Q(\sqrt{−1}) or Q(\sqrt{−3}) is contained in kG and we obtain (9.6). □

Lemmas 9.1, 9.3 and 9.5 yield necessary conditions for the existence of a non-simple elliptic. We conclude this section with a sufficient condition for the existence of a non-simple elliptic.

Lemma 9.7 Suppose that f and h are Möbius transformations and that f is elliptic of order n ≥ 3. If

γ(f, h) ∈ (β(f), 0)

or if

γ(f, h) = β(f) and β(h) ≠ −4,

then the axes of f and hfh^{−1} intersect in a single point.

Proof. Let β = β(f) and γ = γ(f, h). Then

γ(f, hfh^{−1}) = γ(γ − β)  (9.8)

and by Lemma 2.1,

2 cosh(δ(f, hfh^{−1}))^{2} = cosh(2δ(f, hfh^{−1})) + 1
= |\frac{4γ(f, hfh^{−1})}{β(f)β(hfh^{−1})} + 1| + |\frac{4γ(f, hfh^{−1})}{β(f)β(hfh^{−1})}| + 1
= |\frac{4γ(γ − β) + β^{2}}{β^{2}} | + |\frac{4γ(γ − β)}{β^{2}} |
= 2 (\frac{|γ − β| + |γ|}{|β|})^{2}

from which we obtain

cosh(δ(f, hfh^{−1})) = \frac{|γ − β| + |γ|}{|β|}.  (9.9)

If γ ∈ (β, 0), then f and hfh^{−1} have disjoint fixed points by (9.8), δ(f, hfh^{−1}) = 0 by (9.9) and hence axis(f) and axis(hfh^{−1}) intersect in a single point in H^{3}. If γ = β and if β(h) ≠ −4, then (9.8) and the fact that h is not of order 2 imply that f and hfh^{−1} have a single fixed point in common. □

10 Simple axes in extremal groups

Finally, motivated by the search for lower bounds for the volume of hyperbolic 3-orbifolds, we apply the criteria of §9 to determine whether or not the generator f of order n in the groups G_{n,i} is simple. The results are summarized in Table 12 at the end of this section.

Case n=3

From Table 6 in §8 and Lemma 9.1 we can deduce that for i = 6, 7, 10 the groups G_{3,i} do not have subgroups isomorphic to A_{4}, S_{4} or A_{5}. In addition, G_{3,12} is Fuchsian. Hence f is simple in these four cases. For the other cases we apply Lemma 9.7 with β(f) = −3 to show that f is not simple by exhibiting a suitable element h in G_{n,i}.

1. G_{3,1}: γ(f, g) = −1 and ⟨f, g⟩ is isomorphic to S_{4}.

2. G_{3,2}: γ(f, g) = −.3819 … and ⟨f, g⟩ is isomorphic to A_{5}.

3. G_{3,3}: If h = gfg, then γ(f, h) = −2.618 . . ..

4. G_{3,4}: If h = gfgfgfg, then γ(f, h) = −1.

5. G_{3,5}: If h = gfgfg, then γ(f, h) = −2.

6. G_{3,8}: If h = gfgfgf^{−1}gf^{−1}g, then γ(f, h) = −2.

7. G_{3,9}: If h = gfg, then γ(f, h) = −3 and β(h) = −3.

8. G_{3,11}: If h = gfgfgfgfg, then γ(f, h) = −1.

9. G_{3,13}: If h = kfk^{−1}fk where k = gfgfg. then γ(f, h) = −1.

10. G_{3,14}: If h = gfgfgfg, then γ(f, h) = −2.

Case n=4

The methods here are the same as for n = 3. In particular, the cases where i = 2, 5, 6 are dealt with by Lemma 9.1 as above while G_{4,10} is Fuchsian. Hence f is simple in these four cases. Otherwise Lemma 9.7 with β(f) = −2 shows that f is not simple when i ≠ 7, 11.

1. G_{4,1}: γ(f, g) = −1 and ⟨f, g⟩ is isomorphic to S_{4}.

2. G_{4,3}: If h = gfgfg, then γ(f, h) = −1.

3. G_{4,4}: If h = gfg, then γ(f, h) = −2 and β(h) = −2.

4. G_{4,8}: If h = gfgfgfg, then γ(f, h) = −1.

5. G_{4,9}: If h = gfgfg, then γ(f, h) = −2 and β(h) = 2i.

6. G_{4,12}: If h = gfgfgf^{−1}gf^{−1}g, then γ(f, h) = −1.

7. G_{4,13}: If h = gfgfgfgfg, then γ(f, h) = −1.

Case n=5

In this case, the only group which can be dealt with directly by the methods of §9 is G_{5,7}. In this example, although there is only dyadic ramification, the algebra is not isomorphic to

(\frac{−1, −1}{k_{5,7}})

as is shown by Lemma 9.3.

Finally since G_{5,5} is Fuchsian we conclude that f is simple when i = 4, 5, 7. We can then apply Lemma 9.7 with β(f) = −1.3819 … to show that f is not simple when i ≠ 8, 9, 11.

1. G_{5,1}: γ(f, g) = −.38196 … and ⟨f, g⟩ is isomorphic to A_{5}.

2. G_{5,2}: If h = gfg, then γ(f, h) = −1.

3. G_{5,3}: If h = gfgfg, then γ(f, h) = −.38196 . . ..

4. G_{5,6}: If h = gfgfg, then γ(f, h) = −1.

5. G_{5,10}: If h = gfgfgf^{−1}gf^{−1}g, then γ(f, h) = −.38196 . . ..

6. G_{5,12}: If h = gfgfgfg, then γ(f, h) = −1.

Case n=6

When n = 6, Lemma 9.5 implies that the only cases for which f cannot be simple are G_{6,1} and G_{6,3}. The calculations for these using Lemma 9.7 with β(f) = −1 are given below.

1. G_{6,1}: If h = gfg, then γ(f, h) = −1 and β(h) = −1.

2. G_{6,3}: If h = gfgfg, then γ(f, h) = −1 and β(h) = .5 + i2.598 . . ..

The six groups G_{4,7}, G_{4,11}, G_{5,4}, G_{5,8}, G_{5,9} and G_{5,11} have not been treated in the above calculations. The program of Jones and Reid mentioned in §8 shows that f is simple in all of these groups.

We summarize these results in Table 12 below.

Table 12 - f simple elliptic

i G_{3,i} G_{4,i} G_{5,i} G_{6,i}
1 S_{4}  S_{4}  A_{5}  No
2 A_{4}  Yes No Yes
3 No  No  No  No
4 No  No Yes Yes
5 No Yes Fuch. Yes
6 Yes Yes No Yes
7 Yes Yes Yes Fuch.
8 No  No Yes Yes
9 No  No Yes −−
10 Yes Fuch. No −−
11 No Yes Yes −−
12 Fuch. No  No −−
13 No  No −− −−
14 No −− −− −−

11 Commutator parameter diagrams

We conclude this paper with four diagrams which result from the disk covering argument described in §2 for the cases where n = 3, 4, 5, 6. They represent the only possible values for the commutator parameter γ = γ(f, g) of a discrete group ⟨f, g⟩ for which

1. f is an elliptic of order n,

2. γ lies in the union of the indicated disks. The values in these diagrams which correspond to subgroups of arithmetic groups are listed in Tables 1, 2, 3 and 4 in §6. As noted earlier in §2, all of the values in the n = 3 diagram have this property.

Only possible values for commutator parameter when n=3 Only possible values for commutator parameter when n=4 Only possible values for commutator parameter when n=5

46

Only possible values for commutator parameter when n=6

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Authors’ addresses

F. W. Gehring University of Michigan, Ann Arbor, Michigan, USA

C. Maclachlan University of Aberdeen, Aberdeen, Scotland

G. J. Martin University of Auckland, Auckland, New Zealand Australian National University, Canberra, Australia

A. W. Reid University of Cambridge, Cambridge, England