Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 21, 1996, 271–287

THE CHORDAL NORM OF DISCRETE MÖBIUS GROUPS IN SEVERAL DIMENSIONS

Chun Cao

University of Michigan, Department of Mathematics Ann Arbor, MI 48109, U.S.A.; ccao@math.lsa.umich.edu

Abstract. Let d(f, g) = sup\{d(f(z), g(z)) : z ∈ \overline{C}\} where f, g are Möbius transformations and d(z_{1}, z_{2}) denotes the chordal distance between z_{1} , z_{2} in \overline{C}. We show that if ⟨f, g⟩ is a discrete group and if fg ≠ gf, then

max{d(f, id), d(g, id)} ≥ c

where .863 ≤ c ≤ .911 · · ·. We also obtain some higher dimensional analogs by means of Clifford numbers.

1. Introduction

Let GM(n) denote the group of all Möbius transformations of \overline{R}^{n} and M(n) the subgroup of GM(n) consisting of all orientation-preserving Möbius transformations. Stereographic projection p is the mapping from \overline{R}^{n} onto the unit sphere S^{n} in R^{n+1} given by

p(x) = e_{n+1} + \frac{2(x − e_{n+1})}{|x − e_{n+1}|^{2}}

where e_{1} , e_{2}, …, e_{n} is the standard basis for \overline{R}^{n}. The chordal distance between two points x and y in \overline{R}^{n} is defined by

d(x, y) = |p(x) − p(y)|.

The chordal metric on GM(n) is set to be

d(f, g) = sup\{d(f(x), g(x)) : x ∈ \overline{R}^{n}\}.

This metric was considered in [3] and [9]. We call d(f) = d(f, id) the chordal norm of f. Then d(f) measures the maximum chordal derivation of f from the identity, 0 ≤ d(f) ≤ 2 and d(f) = 2 if and only if f maps one point of a pair of antipodal points of \overline{R}^{n} onto the other.

A subgroup G of M(n) is discrete if there exists a positive constant k = k(G) such that d(f, g) ≥ k for each distinct f and g in G. Martin shows that a dimension dependent lower bound can be obtained for chordal norms [14].

1991 Mathematics Subject Classification: Primary 30F40.

Research supported in part by grants from the U.S. National Science Foundation.

1.1. Theorem (Martin). Let f and g be two Möbius transformations of B^{n} generating a discrete subgroup. Then

max\{d(f), d(g)\} ≥ \frac{1}{2\sqrt{16 + n}}

unless ⟨f, g⟩ is an elementary nilpotent group.

For plane Möbius transformations, Gehring and Martin [9] obtained

1.2. Theorem (Gehring–Martin). Suppose that ⟨f, g⟩ is a nonelementary discrete subgroup of M. Then

max{d(f), d(g)} ≥ a

where 2(\sqrt{2} − 1) = 0.828 ⋯ ≤ a ≤ 0.911 · · ·.

By means of Clifford numbers, we show in Section 3 that if f and g generate a discrete nonelementary subgroup of M(n), then

max{d(f), d(g)} ≥ .683,
max{d(f), d(g)} ≥ 1.22,
max{d(f), d(g)} ≥ .816,

if f is hyperbolic,

if f is strictly parabolic,

if f is loxodromic and fixes 0 and ∞.

For plane Möbius transformations, we show that if ⟨f, g⟩ is a discrete subgroup of Möbius transformations of \overline{C} and if fg ≠ gf, then

max{d(f), d(g)} ≥ c_{1},
max{d(f), d(g)} ≥ c_{2},
max{d(f), d(g)} ≥ c_{3},

if f is parabolic, if f is elliptic, if f is loxodromic,

where c_{1} = 1, .937 ≤ c_{2} ≤ 1.12 ⋯, .863 ≤ c_{3} ≤ .911 · · ·. We then apply these results to get some necessary and sufficient conditions for a group to be discrete.

2. Clifford numbers

In [1] Ahlfors shows how a 2×2 matrix with entries in a Clifford algebra may be used to describe a Möbius transformation of \overline{R}^{n}. In this section we will briefly review some material on Clifford numbers that is treated in detail in [1] and [2].

The Clifford algebra C_{n} is the associative algebra over the reals generated by elements i_{1} , i_{2}, …, i_{n} subject to the relations i_{k}^{2} = −1 and i_{h}i_{k} = −i_{k}i_{h}, h ≠ k, and no others. An element of C_{n} is called a Clifford number. Every Clifford number a can be expressed uniquely in the form a = ∑ a_{I}I where a_{I} ∈ R and the sum is over all products I = i_{ν_{1}}i_{ν_{2}} ⋯ i_{ν_{p}} with 1 ≤ i_{ν_{1}} < ⋯ < i_{ν_{p}} ≤ n. The null product is included and identified with the real number i_{0} = 1.

The coefficient of the empty product is denoted by a_{0} and called the real part of a. The sum of all other terms of a is referred to as the imaginary part and we write a = Re(a) + Im(a). We sometimes denote Im(a) by a_{c}. The Euclidean norm of a Clifford number is given by |a|^{2} = ∑ a_{I}^{2} = Re(a)^{2} + |a_{c}|^{2} .

There are three involutions of C_{n}. The major involution a → a′ replaces each i_{k} by −i_{k}. It determines an automorphism of C_{n}: (ab)′ = a′b′, (a + b)′ = a′ + b′. The reversion a → a^{∗} replaces each I = i_{ν_{1}}i_{ν_{2}} ⋯ i_{ν_{p}} with I = i_{ν_{p}} ⋯ i_{ν_{2}}i_{ν_{1}} . It defines an anti-automorphism: (ab)^{∗} = b^{∗}a^{∗}, (a + b)^{∗} = a^{∗} + b^{∗}. The third involution a → ā is a composition: ā = a′^{∗} = a^{∗}′, which is again an antiautomorphism.

We identify R^{n} with the subspace spanned by 1, i_{1} , i_{2}, …, i_{n−1} . Clifford numbers of the form x = x_{0} + x_{1}i_{1} + ⋯ + x_{n−1}i_{n−1} are called vectors. Every non-zero vector x is invertible with x^{−1} = x̄|x|^{−2} . The product of nonzero vectors form a multiplicative group Γ_{n}, known as Clifford group.

A Clifford matrix of dimension n is a matrix g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) which satisfies

the conditions

(1) a, b, c, d ∈ Γ_{n} ∪ {0},

(2) ad^{∗} − bc^{∗} = 1,

(3) ab^{∗}, cd^{∗}, c^{∗}a, d^{∗}b ∈ R^{n}. The set of all Clifford matrices is denoted by SL(2, C_{n}). It is Vahlen’s theorem

that SL(2, C_{n}) form a group whose quotient modulo ± ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) is isomorphic to

M(\overline{R}^{n}), the group of orientation preserving transformations of \overline{R}^{n} [2]. A Clifford matrix in dimension n is also a Clifford matrix in dimension n+1. It automatically extends the corresponding transformation in M(\overline{R}^{n}) to one in M(\overline{R}^{n+1}) given by the same matrix.

The following is the classification of Möbius transformations.

f ∈ GM(n) is elliptic if it has a fixed point in H^{n+1} . Such maps are GM(n+1) conjugate to x → T x with T ∈ O(n).

f ∈ GM(n) is parabolic if it has exactly one fixed point, necessarily in \overline{R}^{n}. Such maps are GM(n) conjugate to x → T x + a with T ∈ O(n), a ∈ R^{n} \ {0}. A parabolic which is conjugate to x → x + a is called strictly parabolic.

f ∈ GM(n) is loxodromic if it has exactly two fixed points, both in \overline{R}^{n}. Such maps are GM(n) conjugate to x → rT x with r > 0, r ≠ 1 and T ∈ O(n). If T = I, then f is called hyperbolic.

A subgroup G of GM(n) is said to be elementary if and only if there exists a finite G-orbit in R^{n+1} .

3. Möbius transformations of \overline{R}^{n}

3.1. Lemma. Suppose that f is not elliptic. Define the map θ: GM(n) → GM(n) by θ(g) = gfg^{−1} . If for some n, θ^{n}(g) and f have the same fixed point set, then g(fix(f)) = fix(f) and hence ⟨f, g⟩ is elementary.

Proof. The result follows from the corresponding statement for SL(2, C) [3, Theorem 5.1.4] and the fact that a parabolic or loxodromic element has at most two fixed points in \overline{R}^{n}.

3.2. Lemma. Suppose that Möbius transformations f and g are represented by

(3.3)  A = ( \begin{matrix}u & 0 \\ 0 & u^{∗}^{−1} \end{matrix} ) ,  B = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ,  ad^{∗} − bc^{∗} = 1.

Suppose that f is loxodromic and that g does not keep the fixed point set of f invariant. If ⟨f, g⟩ is discrete, then

(3.4)  d(f)^{2} ≥ 4(|u| − |u|^{−1})^{2}/(|u| + |u|^{−1})^{2},
(3.5)  d(g)^{2} ≥ 4|bc|/(1 + 2|bc|),
(3.6)  d(f)^{2}(1 + |bc|) ≥ 4/(|u| + |u|^{−1})^{2}.

Proof. Let x = |u|^{−1} . Then

d^{2}(f(x), x) ≥ \frac{4(|f(x)| − |x|)^{2}}{(1 + |f(x)|^{2})(1 + |x|^{2})} = 4(|u| − |u|^{−1})^{2}/(|u| + |u|^{−1})^{2}.

This proves (3.4). For (3.5), we let x_{1} = 0, x_{2} = ∞. Since

\begin{aligned}d(g)^{2} & ≥ d^{2}(g(x_{1}), x_{1}) = \frac{4|bd^{−1}|^{2}}{1 + |bd^{−1}|^{2}} , \\ d(g)^{2} & ≥ d^{2}(g(x_{2}), x_{2}) = \frac{4}{1 + |ac^{−1}|^{2}} , \end{aligned}

we have (4 − d(g)^{2})|bc| ≤ d(g)^{2}|ad|. So (3.5) follows.

The proof of (3.6) requires the discreteness of ⟨f, g⟩. It is essentially due to Hersonsky [11], Friedland and Hersonsky [6], and Waterman [15]. We include a proof for completeness. Consider the Shimizu–Leutbecher sequence

(3.7)  B_{0} = B,  B_{n+1} = B_{n}AB_{n}^{−1}.

The relation (3.7) yields

(3.8)  ( \begin{matrix}a_{n+1} & b_{n+1} \\ c_{n+1} & d_{n+1} \end{matrix} ) = ( \begin{matrix}a_{n}ud_{n}^{∗} − b_{n}u^{∗}^{−1}c_{n}^{∗} & −a_{n}ub_{n}^{∗} + b_{n}u^{∗}^{−1}a_{n}^{∗} \\ c_{n}ud_{n}^{∗} − d_{n}u^{∗}^{−1}c_{n}^{∗} & −c_{n}ub_{n}^{∗} + d_{n}u^{∗}^{−1}a_{n}^{∗} \end{matrix} ) .

We observe that if one of a_{n}, b_{n}, c_{n}, d_{n} is zero, then b_{n+1}c_{n+1} = 0. For each n with a_{n}b_{n}c_{n}d_{n} ≠ 0, we let

x_{n} = a_{n}^{−1}b_{n},  y_{n} = c_{n}^{−1}d_{n},  p_{n} = x_{n}/|ux_{n}|,  q_{n} = y_{n}/|uy_{n}|.

Then

(3.9)  \begin{aligned}|b_{n+1}c_{n+1}| & = |a_{n}b_{n}c_{n}d_{n}||u − x_{n}u^{∗}^{−1}x_{n}^{−1}||u − y_{n}u^{∗}^{−1}y_{n}^{−1}| \\ & = |a_{n}b_{n}c_{n}d_{n}|d(f(p_{n}), p_{n})d(f(q_{n}), q_{n})(|u| + |u|^{−1})^{2}/4 \\ & ≤ |b_{n}c_{n}|(1 + |b_{n}c_{n}|)d(f)^{2}(|u| + |u|^{−1})^{2}/4. \end{aligned}

Suppose that μ = (1 + |bc|)d(f)^{2}(|u| + |u|^{−1})^{2}/4 < 1. We will obtain a contradiction. We obtain, by induction,

|b_{n}c_{n}| ≤ μ^{n}|bc| ≤ |bc|.

So,

b_{n}c_{n} → 0, a_{n}d_{n}^{∗} → 1.

It follows from (3.8) that

|a_{n}| → |u|, |d_{n}| → |u|^{−1}.

Now

|b_{n+1}|/|b_{n}| ≤ |a_{n}|d(f)(|u| + |u|^{−1})/2.

Thus, by induction, |b_{n}|/|u|^{n} → 0, and similarly, |c_{n}||u|^{n} → 0. So

A^{−n}B_{2n}A^{n} = ( \begin{matrix}u^{−n}a_{2n}u^{n} & u^{−n}b_{2n}u^{∗}^{−n} \\ u^{∗}^{n}c_{2n}u^{n} & u^{∗}^{n}d_{2n}u^{∗}^{−n} \end{matrix} )

has a subsequence that converges to a diagonal matrix. Since ⟨f, g⟩ is discrete, u^{−n_{k}}b_{2n_{k}}u^{∗}^{−n_{k}} = 0 and u^{∗}^{n_{k}}c_{2n_{k}}u^{n_{k}} = 0 for sufficiently large k. Thus b_{n} = c_{n} = 0 for infinitely many n. Hence g(fix(f)) = fix(f) by Lemma 3.1. This contradicts the assumption that g does not keep the fixed point set of f invariant. Therefore μ ≥ 1.

3.10. Corollary. Suppose that f is loxodromic with fix(f) = {0, ∞} and that g does not keep the fixed point set of f invariant. If ⟨f, g⟩ is a discrete subgroup of M(n), then

max{d(f), d(g)} ≥ .816.

Proof. Since fix(f) = {0, ∞}, f and g can be represented by the Clifford matrices A and B as in (3.3). Let k = .816, s = 1.543. If |u| ≥ s or |u|^{−1} ≥ s, then d(f) ≥ k by (3.4). If |bc| ≥ \frac{1}{4} , then d(g) ≥ k by (3.5). Finally, if 1/s < |u| < s and |bc| < \frac{1}{4} , then (3.6) implies that d(f) ≥ k.

3.11. Theorem. Suppose that ⟨f, g⟩ is a discrete subgroup of M(n). If f is hyperbolic and g does not keep the fixed point set of f invariant, then

max{d(f), d(g)} ≥ .683.

Proof. The statement is invariant with respect to conjugation by chordal isometries. Thus by means of such a conjugation we may assume that f(∞) = ∞. Thus f and g can be represented by

A = ( \begin{matrix}u & t \\ 0 & u^{∗}^{−1} \end{matrix} ) ,  B = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ,  ad^{∗} − bc^{∗} = 1

where u is real. Replacing f by f^{−1} if necessary, we may assume |u| > 1. Let w be the other fixed point of f. Then w satisfies the equation

(3.12)  uw + t = wu^{∗}^{−1}.

Let h be the Möbius transformation represented by C = ( \begin{matrix}1 & −w \\ 0 & 1 \end{matrix} ). Then

CAC^{−1} = ( \begin{matrix}u & 0 \\ 0 & u^{∗}^{−1} \end{matrix} ) , CBC^{−1} = ( \begin{matrix}a − wc & aw + b − w(cw + d) \\ c & cw + d \end{matrix} ) .

We now consider the sequence

B_{n+1} = B_{n}CAC^{−1}B_{n}^{−1}, B_{0} = CBC^{−1}.

We obtain as in the proof of Lemma 3.2 (see (3.9)) that

|b_{n+1}c_{n+1}| = |a_{n}b_{n}c_{n}d_{n}| |β|

where β = (u − u^{∗}^{−1})^{2} . The same argument as in the proof of Lemma 3.2 yields

(3.13)  |β| + |β(aw + b − w(cw + d))c| ≥ 1.

Since

\begin{aligned}|aw + b|^{2} + |cw + d|^{2} & ≤ ∥g∥^{2}(1 + |w|^{2}), \\ |t|^{2}(4 − d^{2}(f(0), 0)) & ≤ d^{2}(f(0), 0), (|u| > 1) \\ 4|c|^{2} & ≤ ∥g∥^{2}d^{2}(g(∞), ∞), \end{aligned}

it follows from (3.13) and (3.12) that

(3.14)  |β| + \frac{1}{4}(|β| + \frac{d(f)^{2}}{4 − d(f)^{2}} )∥g∥^{2}d(g)^{2} ≥ 1.

Now suppose that

(3.15)  max{d(f), d(g)} < .683.

We will obtain a contradiction. It is a consequence of [7, Theorem 3.3] that

∥g∥^{2} ≤ 2\frac{4 + d(g)^{2}}{4 − d(g)^{2}} .

Hence if |β| ≤ .742, then either d(f) ≥ .683 or d(g) ≥ .683 by (3.14). This contradicts (3.15).

Suppose that |β| > .742. Notice that

4d(f)^{−2} = inf_{x∈\overline{R}^{n}} \frac{|f(x)x̄ + 1|^{2}}{|f(x) − x|^{2}} + 1.

Let x = t/|ut|. We obtain

(3.16)  4d(f)^{−2} − 1 ≤ (|t| + 2)^{2}/|β|.

Since |t|^{2} ≤ d(f)^{2}/(4−d(f)^{2}), (3.16) yields d(f) > .683. This contradicts (3.15).

3.17. Theorem. Suppose that ⟨f, g⟩ is a discrete subgroup of M(n). If f is strictly parabolic and g does not fix the fixed point of f, then

max{d(f), d(g)} ≥ 1.22.

Proof. The statement is invariant with respect to conjugation by chordal isometries. Thus by means of such a conjugation we may assume that f(∞) = ∞. Hence f and g can be represented by

A = ( \begin{matrix}1 & t \\ 0 & 1 \end{matrix} ) ,  B = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ,  ad^{∗} − bc^{∗} = 1.

By [15, Lemma 2.1], |tc| ≥ 1. Since

\begin{aligned}d^{2}(f(0), 0) & = |t|^{2}(4 − d^{2}(f(0), 0)), \\ 4|c|^{2} & ≤ ∥g∥^{2}d^{2}(g(∞), ∞), \\ ∥g∥^{2} & ≤ 2\frac{4 + d(g)^{2}}{4 − d(g)^{2}} , [7, Theorem3.1] \end{aligned}

we have

\frac{d(f)^{2}d(g)^{2}(4 + d(g)^{2})}{(4 − d(f)^{2})(4 − d(g)^{2})} ≥ 2.

Therefore max{d(f), d(g)} ≥ 1.22.

4. Plane Möbius transformations

Let M denote the group of all orientation-preserving Möbius transformations of the extended complex plane \overline{C} = C ∪ \{∞\}. We associate with each

f = \frac{az + b}{cz + d} ∈ M, ad − bc = 1,

the matrix

A = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, C)

and set tr(f) = tr(A), where tr(A) denotes the trace of A. Note that tr(f) is defined up to sign. The matrix norm m(f) is defined by (see [9])

m(f) = ∥A − A^{−1}∥ = (2|a − d|^{2} + 4|b|^{2} + 4|c|^{2})^{1/2}.

The quantity ∥A − A^{−1}∥ is independent of the choice of A representing f and is invariant with respect to conjugation by chordal isometries.

For each f and g in M we let [f, g] denote the multiplicative commutator fgf^{−1}g^{−1} . We call the three complex numbers

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

the parameters of the two generator group ⟨f, g⟩. These parameters are independent of the choice of representative matrices for f and g, and they determine ⟨f, g⟩ up to conjugacy whenever γ(f, g) ≠ 0 [8]. But see [4] for three generator Möbius groups. Note that γ(f, g) ≠ 0 if and only if f and g do not have a common fixed point in \overline{C}.

There are some necessary conditions for a two generator group to be discrete.

4.1. Theorem. If ⟨f, g⟩ is discrete with γ(f, g) ≠ 0 and γ(f, g) ≠ β(f), then

|γ(f, g)| + |β(f)| ≥ 1.

4.2. Theorem. If ⟨f, g⟩ is discrete with γ(f, g) ≠ 0 and γ(f, g) ≠ β(f) and if |β(f)| ≤ 2(cos(2π/7) + cos(π/7) − 1) = 1.0489 …, then

|γ(f, g)| ≥ 2 − 2 cos(π/7) = 0.198 ….

Theorem 4.1 is due to Jørgensen [12]. A proof of Theorem 4.2 is given in [5]. We will quantify the above statements in terms of matrix and chordal norms. See [9] for related results.

4.3. Lemma. Suppose that ⟨f, g⟩ is a discrete subgroup of M with γ(f, g) ≠ 0 and γ(f, g) ≠ β(f). If f is loxodromic, then

(4.4)  m(f)m(g) ≥ 4(2 − 2 cos(π/7))^{1/2} = 1.78 ….

If f is elliptic of order greater than two, then

(4.5)  m(f)m(g) ≥ 4(2 cos(2π/7) − 1)^{1/2} = 1.987 ….

Inequality (4.5) is sharp.

Proof. Let c_{0} = 2(cos(2π/7) + cos(π/7) − 1), d_{0} = 2 − 2 cos(π/7). Suppose that f is loxodromic. If |β(f)| ≥ c_{0} and |β(g)| ≥ c_{0} , then

m(f)m(g) ≥ 2|β(f)β(g)|^{1/2} ≥ 2c_{0} = 2.097 …

by [9, Theorem 2.7]. If |β(g)| ≤ c_{0} , then |γ(f, g)| ≥ d_{0} by [5, Theorem 3.1]. If |β(f)| ≤ c_{0} , then |γ(f, g)| ≥ d_{0} by Theorem 4.2. Thus by [9, Theorem 2.7],

m(f)m(g) ≥ 4|γ(f, g)|^{1/2} ≥ 4\sqrt{d_{0}} = 1.78 ….

Suppose next that f is elliptic of order greater than two. Then by [10, Theorem 3.1], |γ(f, g)| ≥ 2 cos(2π/7) − 1. Therefore,

m(f)m(g) ≥ 4(2 cos(2π/7) − 1)^{1/2}.

The (2, 3, 7) triangle group in [9, Lemma 4.8] shows that (4.5) is sharp.

4.6. Remark. (i) Gehring and Martin [9] have shown that if ⟨f, g⟩ is nonelementary discrete, then

m(f)m(g) ≥ 4(\sqrt{2} − 1) = 1.656 ….

Furthermore, m(f)m(g) ≥ 4 if f is parabolic.

(ii) Let f, g be the Möbius transformations represented by

A = ( \begin{matrix}cos(π/n) & i sin(π/n) \\ i sin(π/n) & cos(π/n) \end{matrix} ) , B = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) .

Then ⟨f, g⟩ is discrete and γ(f, g) = β(f) = −4 sin^{2}(π/n). In this case,

m(f)m(g) = 8 sin(π/n) → 0 as n → ∞.

Notice that γ(g, f) ≠ β(g) and g is elliptic of order two.

(iii) If f is of order two and γ(f, g) ≠ 0, γ(f, g) ≠ β(g), then m(f)m(g) ≥ 2\sqrt{2}. This is because that if |β(g)| ≥ 1/2, then m(f)m(g) ≥ 2|β(f)β(g)|^{1/2} ≥ \sqrt{8}. If |β(g)| ≤ 1/2, then |γ(f, g)| ≥ 1/2 by Jørgensen’s inequality. So m(f)m(g) ≥ 4|γ(f, g)|^{1/2} ≥ \sqrt{8}.

4.7. Lemma. Let e(c) = inf{d(f) : fis elliptic and m(f) = c}. Then

(4.8)  e(c) = \{ \begin{matrix}\frac{c}{\sqrt{2}} & if 0 < c ≤ 4(\sqrt{2} − 1)^{1/2}, \\ \frac{16c}{c^{2} + 16} & if 4(\sqrt{2} − 1)^{1/2} < c < 4, \\ 2 & if 4 ≤ c. \end{matrix}

Proof. All quantities in (4.8) are invariant with respect to conjugation by chordal isometries. Thus by means of such a conjugation we may arrange that fix(f) = {−r, r} where 0 < r ≤ 1. Since f is elliptic, it is conjugate to a mapping of the form w = e^{i2θ}z, −π/2 < θ ≤ π/2. By [9, Lemma 2.10],

m(f)^{2} = 2(8 − q(−r, r)^{2})q(−r, r)^{−2}|β(f)|
= (r^{2} + r^{−2})|e^{iθ} − e^{−iθ}|^{2}
= (r^{2} + r^{−2})4 sin^{2} θ.

Note that m(f) ≥ 4 if and only if | tan(θ/2)| ≥ r. Thus by [9, Lemma 3.1], d(f) = 2 if c ≥ 4. We now assume that | tan(θ/2)| < r. Then [9, Lemma 3.1] implies that

16d(f)^{−2} − 2 = r^{2} cot^{2}(θ/2) + r^{−2} tan^{2}(θ/2).

We will find the maximum value of

g(r, θ) = r^{2} cot^{2}(θ/2) + r^{−2} tan^{2}(θ/2), 0 < r ≤ 1,

subject to the constraint

h(r, θ) = 4(r^{2} + r^{−2}) sin^{2} θ − c^{2} = 0.

Let F (r, θ) = g(r, θ) + λh(r, θ). Then the critical points satisfy the equations:

\frac{∂F}{∂r} = 0, \frac{∂F}{∂θ} = 0.

It follows that

(4.9)  r^{2} cot^{2}(θ/2) − r^{−2} tan^{2}(θ/2) = 4λ(r^{−2} − r^{2}) sin^{2} θ,
(4.10)  r^{2} cot^{2}(θ/2) − r^{−2} tan^{2}(θ/2) = 4λ(r^{−2} + r^{2}) sin^{2} θ cos θ.

Since | tan(θ/2)| < r, (4.9)/(4.10) gives cos θ = (1 − r^{4})(1 + r^{4})^{−1} . Hence at the critical points,

(4.11)  g(r, θ) = 16/c^{2}.

Next we consider the case r → 0. Solving θ from the equation h(r, θ) = 0, we get

(4.12)  lim_{r→0} g(r, θ) = lim_{r→0}(r^{2} cot^{2}(θ/2) + r^{−2} tan^{2}(θ/2)) = 16/c^{2} + c^{2}/16.

At the end point r = 1, sin^{2} θ = c^{2}/8. Hence

(4.13)  g(1, θ) = 32/c^{2} − 2.

Combining (4.11), (4.12) and (4.13), we obtain (4.8).

4.14. Remark. Let p(c) = inf{d(f) : f is parabolic and m(f) = c} and let l(c) = inf{d(f) : f is loxodromic and m(f) = c}. Then

l(c) = \frac{2c}{(c^{2} + 8)^{1/2}} , p(c) = \{ \begin{matrix}\frac{16c}{c^{2} + 16} & if 0 < c < 4, \\ 2 & if 4 ≤ c \end{matrix}

by [9, Theorem 3.11 and Lemma 3.8]. It is easy to check that p(c), e(c), l(c) are continuous increasing functions of c and p(c) ≥ e(c) > l(c).

4.15. Lemma. Suppose that ⟨f, g⟩ is a discrete subgroup of M and that f is elliptic. If fg ≠ gf, then

max{d(f), d(g)} ≥ c_{1}, .937 ≤ c_{1} ≤ 1.121 ….

Proof. Suppose first that γ(f, g) ≠ 0 and γ(f, g) ≠ β(f). If f is of order two, then d(f) = 2 by [9, Corollary 3.17]. Suppose that f is elliptic of order greater than two. Let a_{0} = 4(2 cos(2π/7) − 1)^{1/2} , t = m(f). Then m(g) ≥ a_{0}/t by Lemma 4.3. By Remark 4.14,

max{d(f), d(g)} ≥ max{e(m(f)), l(m(g))} ≥ max{e(t), l(a_{0}/t)}.

Note that e(x) is strictly increasing for x ≤ 4 and l(a_{0}/x) is strictly decreasing. Hence max{e(x), l(a_{0}/x)} obtains its minimum when x is the intersection point x_{0} of e(x) and l(a_{0}/x). Solving for x, we get

\begin{aligned}x_{0} & = ((a_{0}^{2} + (a_{0}/4)^{4})^{1/2} − (a_{0}/4)^{2})^{1/2}, \\ max_{x>0}\{e(x), l(a_{0}/x)\} & ≥ x_{0}/\sqrt{2} = .937 …. \end{aligned}

Suppose next that γ(f, g) ≠ 0, γ(f, g) = β(f). Then either f is elliptic of order 2, 3, 4, or 6 or g is elliptic of order 2 by [10, Lemma 2.31]. In either case,

max{d(f), d(g)} ≥ 2 sin(π/6) = 1

by [9, Corollary 3.17]. Finally, suppose that γ(f, g) = 0. Then f and g have a common fixed point in \overline{C}, say ∞. Thus every element of ⟨f, g⟩ fixes ∞. Since fg ≠ gf, [f, g] is parabolic by [3, Theorem 4.3.5]. So there are no loxodromic elements in ⟨f, g⟩ by [3, Theorem 5.1.2]. By the structure of elementary groups [3, § 5.1], if S is the set of multipliers of ⟨f, g⟩, then S = {1, ω, ω^{2}, …, ω^{q−1}} where ω = exp(2πi/q), 0 ≤ q ≤ 6, q ≠ 5. So f is elliptic of order less than or equal to six. Therefore,

max{d(f), d(g)} ≥ d(f) ≥ 2 sin(π/6) = 1.

To get the number 1.121 …, let ⟨ϕ, ψ⟩ denote the (2, 3, 7) triangle group with ϕ^{2} = ψ^{3} = (ϕψ)^{7} = id. The transformations ϕ and ψ can be represented by the matrices

A = \frac{i}{sin a} ( \begin{matrix}− cos b & −p \\ p & cos b \end{matrix} ) , B = ( \begin{matrix}e^{ia} & 0 \\ 0 & e^{−ia} \end{matrix} )

where a = π/3, b = π/7 and p = (cos^{2} b−sin^{2} a)^{1/2} [13, p. 88]. We set f = [A, B] and g = AB. Then

β(f) = 2(cos(2π/7) + cos(π/7) − 1),  β(g) = 2 cos(2π/7) − 2,
γ(f, g) = 2 cos(2π/7) − 1.

We can find a Möbius transformation h which sends the fixed points of f to {w, −w} and sends the fixed points of g to {1/w, −1/w}. By [9, Lemma 2.12], such a w satisfies the equation

(w^{2} − 1/w^{2})^{2} = 16\frac{γ(f, g)}{β(f)β(g)}.

Let u = |w|^{2} + 1/|w|^{2} . Then m^{2}(hfh^{−1}) = uβ(f) and m^{2}(hgh^{−1}) = −uβ(g) by [9, Lemma 2.10]. It is a consequence of [9, Lemma 3.1] that for any Möbius transformation f, if m(f)^{2} ≤ 2(|β + 4| + 4), then

\begin{aligned}d(f) & = \frac{2(|β + 4| + 4 + |β|)(\frac{1}{2}m(f)^{2} + |β|))^{1/2}}{|β + 4| + 4 + \frac{1}{2}m(f)^{2}} \\ & + \frac{2(|β + 4| + 4 − |β|)(\frac{1}{2}m(f)^{2} − |β|))^{1/2}}{|β + 4| + 4 + \frac{1}{2}m(f)^{2}} . \end{aligned}

Therefore

d(hfh^{−1}) = 1.121 …, d(hgh^{−1}) = 1.071 ….

4.16. Example. Let f = e^{2πi/m}z, g = e^{2πi/n}z. Then ⟨f, g⟩ is a discrete finite group with fg = gf. We have d(f) → 0, d(g) → 0, as m, n → ∞.

4.17. Lemma. Suppose that ⟨f, g⟩ is a discrete subgroup of M and that f is parabolic. If fg ≠ gf, then

max{d(f), d(g)} ≥ 1.

Proof. If γ(f, g) ≠ 0, then m(f)m(g) ≥ 4 by [9, Lemma 4.5]. Let t = m(f). By Remark 4.14,

max{d(f), d(g)} ≥ max{p(m(f)), l(m(g))} ≥ max{p(t), l(4/t)}.

Note that p(x) is strictly increasing for x ≤ 4 and l(4/x) is strictly decreasing. Hence max{p(x), l(4/x)} obtains its minimum when x is the intersection point x_{0} of p(x) and l(4/x). Solving for x_{0} , we have

\begin{aligned}x_{0} & = \frac{4}{31}(124\sqrt{2} − 31)^{1/2}, \\ max_{x>0}\{p(x), l(4/x)\} & ≥ 4/(4 + 2x_{0}^{2})^{1/2} = 1.347 …. \end{aligned}

So,

max{d(f), d(g)} ≥ 1.347 ….

If γ(f, g) = 0, then f and g have a common fixed point in \overline{C}, say ∞. Thus every element of ⟨f, g⟩ fixes ∞. Since f is parabolic, there are no loxodromic elements in ⟨f, g⟩ by [3, Theorem 5.1.2]. Since fg ≠ gf, g is not parabolic by [3, Theorem 4.3.6]. Thus g is elliptic. By the structure of elementary groups [3, § 5.1], the set of multipliers S = {1, ω, ω^{2}, …, ω^{q−1}} where ω = exp(2πi/q), 0 ≤ q ≤ 6, q ≠ 5. So g is elliptic of order less than or equal to six. Therefore,

max{d(f), d(g)} ≥ d(g) ≥ 2 sin(π/6) = 1.

4.18. Corollary. Suppose that ⟨f, g⟩ is a discrete subgroup of M and that f is parabolic. If γ(f, g) ≠ 0, then

max{d(f), d(g)} ≥ c_{2}, 1.347 ≤ c_{2} ≤ 1.6.

Proof. It follows from the proof of Lemma 4.17 that max{d(f), d(g)} ≥ 1.347. The subgroup generated by

f = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) , g = ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} )

is discrete. We have d(f) = d(g) = 8/5.

4.19. Example. Let f = z + 1/m, g = z + 1/n. Then ⟨f, g⟩ is a discrete group with fg = gf. We have d(f) → 0, d(g) → 0, as m, n → ∞.

4.20. Lemma. Suppose that ⟨f, g⟩ is a discrete subgroup of M and that f is loxodromic. If fg ≠ gf, then

max{d(f), d(g)} ≥ c_{3}, .863 ≤ c_{3} ≤ .911 ….

Proof. By Lemma 4.15 and Lemma 4.17, we may assume that g is also loxodromic. Thus γ(f, g) ≠ β(f) by [10, Lemma 2.31]. We also have γ(f, g) ≠ 0 since otherwise f and g have one common fixed point, and hence two common fixed points by [3, Theorem 5.1.2]. Thus fg = gf, a contradiction.

Since f is loxodramic, it is conjugate to a mapping of the form w = ρ^{2}e^{i2θ}z, −π/2 < θ ≤ π/2, 0 < ρ ≠ 1.

By [9, Theorem 2.7 and Theorem 3.11],

(4.21)  2|β(f)| ≤ m(f)^{2} ≤ 8 cos^{2} θ \frac{d(f)^{2}}{4 − d(f)^{2}} .

Let c_{0} = 2(cos(2π/7) + cos(π/7) − 1), d_{0} = 2 − 2 cos(π/7). It follows from (4.21) that if |β(f)| ≥ c_{0} , then

d(f) ≥ 2(\frac{cos(2π/7) + cos(π/7) − 1}{cos(2π/7) + cos(π/7) + 1})^{1/2} = .911 ….

Suppose that max{|β(f)|, |β(g)|} ≤ c_{0} . Let γ = γ(f, g), β = β(f), v = cos^{2} θ, a = 0.21. If |γ| ≥ a, then by [9, Theorem 2.7 and Corollary 3.15],

max\{d(f), d(g)\} ≥ (\frac{4|γ|^{1/2}}{|γ|^{1/2} + 2})^{1/2} ≥ .863.

We now assume that |γ| ≤ a. Then |β| ≥ 1 − a by Jørgensen’s inequality (4.1). Let g_{1} = gf^{−1}g^{−1}fgfg^{−1}f^{−1}g. If γ(f, g_{1}) ≠ 0 and γ(f, g_{1}) ≠ β(f), then

|γ(f, g_{1})| = |γ(γ^{2} − (β − 1)γ − (β − 1))^{2}| ≥ d_{0}

by [5, Corollary 3.8]. It follows that

(4.22)  |β − 1| > \frac{1}{1 + a}(\sqrt{d_{0}/a} − a^{2}).

Since 1 − a ≤ |β| ≤ c_{0} and

|β − 1|^{2} = −16v^{2} + 4(4 − |β|)v + (|β| + 1)^{2},

it follows from (4.22) that v < .971. If m(f) ≥ m(g), then

1.78 ≤ m(f)m(g) ≤ m(f)^{2} ≤ \frac{8vd(f)^{2}}{4 − d(f)^{2}}

by Lemma 4.3 and (4.21). Therefore,

max{d(f), d(g)} ≥ d(f) ≥ .863.

If γ^{2} − (β − 1)γ − (β − 1) = 0, then β = (1 + γ + γ^{2})/(1 + γ). Since |γ| ≤ a,

\frac{1}{|β|} ≤ max_{|z|=a}|\frac{1 + z}{1 + z + z^{2}} | < 1.05.

It follows from (4.21) that d(f) > .877.

We finally show that γ(f, g_{1}) = β(f) can not occur. By [10, Lemma 2.29], there exists an elliptic h of order two such that ⟨f, h⟩ is discrete with γ(f, h) = γ(f, g). Let g_{2} = hf^{−1}h^{−1}fhfh^{−1}f^{−1}h. Suppose that γ(f, g_{1}) = β(f). We will obtain a contradiction. Since γ(f, g_{2}) = γ(f, g_{1}) = β(f), g_{2} is of order two by [10, Lemma 2.31]. After a conjugation, we may assume that f and h are represented by

A = ( \begin{matrix}u & 0 \\ 0 & 1/u \end{matrix} ) , B = ( \begin{matrix}e_{11} & e_{12} \\ e_{21} & e_{22} \end{matrix} ) .

Thus β = (u − 1/u)^{2} , γ = −e_{12}e_{21}(u − 1/u)^{2} . Elementary calculations show that

\begin{aligned}BA^{−1} & B^{−1}ABAB^{−1}A^{−1}B = \\ & ( \begin{matrix}e_{11}((γ + 1)^{2} − γu^{−2}) & e_{12}(γ^{2} − (β − 1)γ − (β − 1)) \\ e_{21}(γ^{2} − (β − 1)γ − (β − 1)) & e_{22}((γ + 1)^{2} − γu^{2}) \end{matrix} ) . \end{aligned}

Since hf^{−1}h^{−1}fhfh^{−1}f^{−1}h is of order two,

(4.23)  e_{11}((γ + 1)^{2} − γu^{−2}) + e_{22}((γ + 1)^{2} − γu^{2}) = 0.

Notice that e_{11} + e_{22} = 0 (h is of order two). If e_{11} ≠ 0, then (4.23) implies that β(β + 4) = 0, a contradiction. If e_{11} = 0, then e_{12}e_{21} = −1. Hence γ = β, another contradiction.

The number .911 … occurs in the (2, 3, 7) triangle group (see [9]).

4.24. Example. Let f = (1+1/m)z, g = (1+1/n)z. Then ⟨f, g⟩ is discrete and fg = gf. We have d(f) → 0, d(g) → 0, as m, n → ∞.

4.25. Theorem. Suppose that ⟨f, g⟩ is a discrete subgroup of M. If fg ≠ gf, then

max{d(f), d(g)} ≥ c, .863 ≤ c ≤ .911 ….

Proof. This follows from Lemma 4.15, Lemma 4.17 and Lemma 4.20.

4.26. Corollary. Suppose that ⟨f, g⟩ is a discrete subgroup of M. If γ(f, g) ≠ 0 and g is not of order two, then

max{d(f, g), d(f, g^{−1})} ≥ .863.

Proof. This follows from the fact d(f, g) = d(fg^{−1}) and Theorem 4.25.

4.27. Theorem. Suppose that G does not have a G-orbit in \overline{C} that has less than three points. Then G is discrete if and only if for each pair f, g ∈ G \ {id}, either fg ≠ gf or

max{d(f), d(g)} ≥ .863.

Proof. By Theorem 4.25, it suffices to prove that G is discrete if for each pair f, g ∈ G \ {id}, either max{d(f), d(g)} ≥ .863 or fg = gf. Suppose that G is not discrete. We will obtain a contradiction. Since G is not discrete, there exists distinct elements f_{1} , f_{2} , ⋯ (≠ id) in G such that d(f_{n}, id) → 0 as n → ∞. So there exists an N > 0 such that d(f_{n}, id) < 1/2 if n ≥ N. Thus for all m, k ≥ N, f_{m}f_{k} = f_{k}f_{m} by hypothesis. So we may assume that there exists a sequence {f_{n}} such that

d(f_{n}, id) → 0,_{f_{m}f_{k} = f_{k}f_{m},} as n → ∞

for all m, k ≥ 1.

Furthermore, since all f_{i} ’s are distinct and d(f_{n}, id) → 0, by passing to a subsequence of {f_{n}} if necessary, we may assume that f_{k} is not of order two for all k ≥ 1. Since f_{k} and f_{m} commute for all k, m ≥ 1, fix(f_{k}) = fix(f_{m}) by [3, Theorem 4.3.6]. Let fix(f_{k}) = {a, b} for all k ≥ 1 (it is possible that a = b). For any g ∈ G \ {id}, d(gf_{n}g^{−1}) → 0, as n → ∞. So there exists an n_{0} such that max{d(gf_{n}g^{−1}), d(f_{n})} ≤ 1/2 if n ≥ n_{0} . By hypothesis, gf_{n_{0}}g^{−1}f_{n_{0}} = f_{n_{0}}gf_{n_{0}}g^{−1} . Since f_{n_{0}} is not of order two, fix(gf_{n_{0}}g^{−1}) = fix(f_{n_{0}}) = {a, b} by [3, Theorem 4.3.6]. Thus g{a, b} = {a, b}. Since g is arbitrary, ∩_{h∈G}h(a) ⊂ {a, b}. This contradicts the assumption that G does not have a G-orbit that has less than three points.

Acknowledgment. The author wishes to thank Professor F.W. Gehring for suggesting this topic and for many helpful conversations. He also wishes to thank the University of Auckland for its hospitality during part of this work and the referee for the careful reading of the manuscript.

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Received 7 March 1995