Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 40 (1999), No. 2, 351-361.

Galleries in Poincaré Half-spaces and Diophantine Approximation

Pierrick Meignen

Structures Discretes et Analyse Diophantienne, U.F.R. des Sciences Université de Caen, Esplanade de la Paix, 14032 Caen, France e-mail: meignen@math.unicaen.fr

Abstract. In this paper, we present a multidimensional generalization of a geometrical method of approximation devised by G. Humbert. Introducing Clifford algebras and hyperbolic reflection groups with unbounded fundamental domains of finite volume, we obtain a complete set of results which are traditionally connected with the modular group and the usual continued fraction expansions in ℝ. Keywords: Hyperbolic reflection groups, Diophantine approximation

1. Introduction

For z in ℂ \ ℚ(i), C. Hermite (1854) described a way to get an infinite set of (p, q) in the ring of the Gaussian integers ℤ[i] such that

|z − \frac{p}{q}| ≤ \frac{1}{\sqrt{2}|q|^{2}} ,  (1)

using method of reduction of binary forms. Following Humbert’s works (1916), L. R. Ford (1917) gave a geometric interpretation by considering the action of the Picard group P SL_{2}(ℤ[i]) on the Poincaré upper half-space of ℝ^{3}. Developing his own methods, A. L. Schmidt (1969) got a similar result in the quaternionic framework of the Hurwitz integers. More recently, L. Ya. Vulakh (1995) obtained this kind of inequality by considering the action of geometrically finite groups (acting on the Poincaré half-space) as a multidimensional generalization of the Humbert-Ford approach. Independently, the author chose in his thesis to restrict himself to hyperbolic reflection groups and got some results that we present in part here.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1999 Heldermann Verlag

352 P. Meignen: Galleries in Poincaré half-spaces …

2. Hyperbolic and Euclidean reflection groups

On ℝ^{p}, with p ≥ 3, we consider the Lorentz quadratic form B of signature (p − 1, 1) and we choose a basis (ϵ_{i}) such that for all x = ∑_{i} x_{i}ϵ_{i} we have

B(x, x) = ∑_{i=1}^{p−1} x_{i}^{2} − x_{p}^{2}.

Let {e_{i}} be a finite set of unitary vectors such that B(e_{i}, e_{j}) = − cos(π/m_{i,j}) for some integer m_{i,j} < ∞, or B(e_{i}, e_{j}) ≤ −1. Let A_{i} = {x ∈ ℝ^{n}|B(x, e_{i}) ≥ 0}. The set S of the orthogonal reflections s_{i} in the walls ∂A_{i} generates a discrete subgroup W_{S} of the Lorentz orthochronous group O^{+}(p − 1, 1) formed by the orthogonal transformations which preserve the positive cone

C = {x ∈ ℝ^{n}|B(x, x) < 0, 0 < x_{n}}.

The relations s_{i}^{2} = 1 and (s_{i}s_{j})^{m_{i,j}} = 1 are defining relations for W_{S} which is a Coxeter group. For w in W_{S}, the length l_{S}(w) is the smallest integer q such that w is the product of q elements of S.

Replacing eventually e_{i} by −e_{i}, we can suppose that ∩_{i}A_{i} has a non-empty intersection C_{S} with the closure \overline{C} of C. For the action of W_{S} on C, C_{S} is a fundamental domain, i.e. the union of the chambers w(C_{S}) for w in W_{S} covers C, and if x and w(x) are in C_{S} then w(x) = x. Furthermore, the isotropy group of any x is W_{S(x)} where S(x) = {s ∈ S|s(x) = x}. We know [3] that for s_{i} ∈ S, we have w(C_{S}) ⊂ A_{i} or w(C_{S}) ⊂ s_{i}(A_{i}) and

w(C_{S}) ⊂ A_{i} ⇔ l_{S}(s_{i}w) = l_{S}(w) + 1.  (2)

We denote by B^{p−1} the Euclidean unit (p−1)-ball with centre ϵ_{p} which is the intersection of the affine hyperplane P = \{x ∈ ℝ^{p}|x_{p} = 1\} with \overline{C}. We shall work with the Poincaré half-space H^{p−1} := {x ∈ P |x_{p−1} > 0}. The stereographic projection S^{p−2} := ∂B^{p−1} → ∂H^{p−1} ∪ {∞} from the north pole v_{+} = ϵ_{p} + ϵ_{p−1} admits a natural extension Φ : B^{p−1} → \overline{H}^{p−1} ∪ \{∞\} by mapping a segment [x_{1}, x_{2}], with x_{1} and x_{2} in S^{p−2}, on a half-line or a halfcircle perpendicular to ∂H^{p−1} ≈ ℝ^{p−2}. This establishes a one-to-one correspondence between geodesics of the affine Klein model for the Euclidean norm ∥ ∥ and geodesics of the Poincaré model for the hyperbolic distance ρ.

For all f in O^{+}(n − 1, 1) the following diagram commutes

B^{n−1} ⟶^{Φ} \overline{H}^{n−1} ∪ \{∞\}
pf↓  ↓^{f̃:=ΦpfΦ^{−1}}
B^{n−1} ⟶^{Φ} \overline{H}^{n−1} ∪ \{∞\}

where p(x) = x/x_{n}. The map f ↦ f̃ is an isomorphism from O^{+}(p − 1, 1) onto the Möbius group Möb(H^{p−1}) acting on the Poincaré half-space.

For s_{i} in S, the corresponding Möbius transformation s̃_{i} is a reflection in hyperplane or an inversion in a half-sphere orthogonal to ∂H^{p−1}. The chamber C̃_{S} := Φ(C_{S}) becomes a fundamental domain for the action of W_{S} on the Poincaré half-space.

If the set Fix(W_{S}) of points fixed by W_{S} in \overline{H}^{p−1} ∪ \{∞\} is empty then we say that W_{S} is an hyperbolic reflection group.

If Fix(W_{S}) ≠ ∅ then W_{S} is conjugate to a discrete subgroup of Euclidean affine motions and we say that W_{S} is a Euclidean reflection group. Two cases may occur: Fix(W_{S}) lies in ∂H^{p−1} ∪ {∞} (W_{S} is infinite) or intersects H^{p−1} (W_{S} is finite).

In [10], Vinberg uses the projective Klein model C/ℝ_{+}^{∗} and the Lobachevskii space model

Λ^{p} := {x ∈ ℝ^{p}|B(x, x) = −1, x_{p} > 0}.

Here we work with three other models for W_{S}: the linear model (ℝ^{p}, B), the affine Klein model (B^{p−1}, ∥ ∥) and the Poincaré model (H^{p−1}, ρ).

3. The Cliffordian framework

We consider the Clifford algebra (Cl_{p−2}, −∥ ∥^{2}) with dimension 2^{p−2} generated by ϵ_{1}, ⋯, ϵ_{p−2} such that

\{ \begin{matrix}ϵ_{i}^{2} = −1 \\ ϵ_{i}ϵ_{j} + ϵ_{j}ϵ_{i} = 0 for i ≠ j \end{matrix}

By mapping ϵ_{0} on the unit 1 of Cl_{p−2}, ∂H^{p−1} may be viewed as the subspace ℝ.1⊕V ⊂ Cl_{p−2} where V = ⊕_{i=1}^{p−1}ℝϵ_{i}. The corresponding elements of ∂H^{p−1} in Cl_{p−2} will be termed vectors. We have Cl_{0} = ℝ, Cl_{1} = ℂ and Cl_{2} = ℍ.

We give the definitions of three involutions of Cl_{p−2}:

• The main automorphism x ↦ x′ defined by ϵ′_{i} = −ϵ_{i} for 2≤ i ≤ p − 1

• The main anti-automorphism x ↦ x^{∗} defined by ϵ_{i}^{∗} = ϵ_{i}

• The anti-automorphism x ↦ \overline{x} := x′^{∗} called conjugation.

The set of products of non-zero vectors of ℝ^{p−2} form the Clifford group denoted by Cg_{p−3}. We know [1] that a map f̃: H^{p−1} → H^{p−1} is an homography (even product of inversions) of Möb(H^{p−1}) if and only if there exist a, b, c and d in Cg_{p−3} ∪ {0} such that

\begin{matrix}i) & ab^{∗}, cd^{∗} ∈ ℝ^{p−2} \\ ii) & ad^{∗} − bc^{∗} = 1 \\ iii) & f̃(x) = (ax + b)(cx + d)^{−1} \end{matrix}

For the anti-homographies, ii) and iii) are replaced by

\begin{matrix}ii)’ & ad^{∗} − bc^{∗} = −1 \\ iii)’ & f̃(x) = (ax′ + b)(cx′ + d)^{−1} \end{matrix}

It is convenient to write f̃ using Cliffordian matrices

( \begin{matrix}a & b \\ c & d \end{matrix} ) or ( \begin{matrix}a & b \\ c & d \end{matrix} )′

according that it is an homography or an anti-homography. We have to note that (a, b, c, d) is unique up to a multiplication by ±1.

For x in Cl_{p−2}, we denote by ℜ(x) the coordinate on the subspace ℝ.1 (the component of degree 0). We denote by | | the norm on Cl_{p−2} which corresponds to the Euclidean structure given by

(x, y) ↦ ℜ(x\overline{y}).

In particular, for x in Cg_{p−3} we have |x|^{2} = x\overline{x}.

For y ∈ \overline{H}^{p−1}, we consider the “form” F_{y} defined on Cl_{p−2} × Cl_{p−2} by

F_{y}(p, q) = |p − yq|^{2}.

We use the same letter to write the Möbius transformation and the corresponding motion in O^{+}(p − 1, 1). For f ∈ Möb(H^{p−1}), we have (see (9) in [8])

F_{y}f(1, 0) = −\frac{1}{2}(1 + |y|^{2})B(f(v_{+}), Φ^{−1}(y)).  (3)

A point x in ∂H^{p−1} ∪ {∞} is said to be parabolic if its isotropy group W_{S(x)} is conjugate to an infinite affine Euclidean group. From now on, we suppose that C̃_{S} is the convex closure of a finite number of points (vertices), and that at least one, say α, is parabolic. Our aim is to approximate a given point in ∂H^{p−1} by some elements in the orbit W_{S}(α) of α. For w in W_{S}, pq^{−1} = w(α) will mean that (p, q) = w(1, 0) or w(α, 1) for α ≠ ∞ using the Cliffordian matrix of w.

Theorem 1. Let α be a parabolic vertex of C̃_{S} and y be in H^{p−1}. The minimum of F_{y} on W_{S}(α) is attained at the parabolic point of the chambers which contain y.

Proof. First, we suppose that y ∈ C̃_{S}. From (3), one can show that

F_{y}ws_{i}(α) − F_{y}w(α) = −2\frac{B(Φ^{−1}(α), e_{i})B(w(e_{i}), Φ^{−1}(y))}{B(v_{+}, Φ^{−1}(y))}  (4)

for any s_{i} ∈ S and w ∈ W_{S}. We have B(v_{+}, Φ^{−1}(y)) < 0 because Φ^{−1}(y) ∈ \overline{C}. We know that B(Φ^{−1}(α), e_{i}) ≥ 0 because Φ^{−1}(α) ∈ C_{S}. From (2) and the fact that Φ^{−1}(y) ∈ C_{S}, we obtain

l_{S}(ws_{i}) > l_{S}(w) ⇒ F_{y}ws_{i}(α) ≥ F_{y}w(α).
Suppose that y ∈ w(C̃_{S}). We use
F_{y}wf(α) = F_{w^{−1}(y)}f(α) × F_{y}w(1, 0)

which holds for all f in W_{S}. We have w^{−1}(y) ∈ C̃_{S} hence

inf{F_{y}(p, q)} = F_{w^{−1}(y)}(α) × F_{y}w(1, 0) = F_{y}w(α).  □

We say that pq^{−1} ∈ W_{S}(α) is a convergent for z ∈ ∂H^{p−1} if it satisfies the best approximation condition

\{ \begin{matrix}|P z − Q| ≤ |pz − q| \\ |Q| ≤ |q| \end{matrix} ⇒ \{ \begin{matrix}|P z − Q| = |pz − q| \\ |Q| = |q| \end{matrix}

for P Q^{−1} ∈ W_{S}(α). From Theorem 1, we get Corollary 1. Let z be in ∂H^{p−1}. A parabolic vertex of a chamber which intersects (z, ∞) is a convergent for z.

Proof. Indeed, for y ∈ (z, ∞) in the chamber, |P z − Q| ≤ |pz − q| and |Q| ≤ |q| implies F_{y}(P, Q) ≤ F_{y}(p, q), hence F_{y}(P, Q) = F_{y}(p, q).

We denote by B_{z}(r) the Euclidean ball in \overline{H}^{p−1} tangent to ∂H^{p−1} at z, with radius r > 0. We also write B_{∞}(r) for {x ∈ P |x_{p−1} ≥ 1/2r} ∪ {∞}.

Corollary 2. For each parabolic vertex α ∈ C̃_{S}, let r_{α} > 0 be such that C̃_{S} is in ∪_{α}B_{α}(r_{α}). Let y be in H^{p−1}. The minimum of F_{y} over ∪_{α}W_{S}(α) is smaller than 2y_{p−1}/γ_{1} where γ_{1} = 1/ sup{r_{α}}.

Proof. There are w ∈ W_{S} and a parabolic vertex α ∈ C̃_{S} such that y is in the ball w(B_{α}(r_{α})) = B_{w(α)}(r_{α}/|q|^{2}) where pq^{−1} = w(α). Let c be the centre of this ball and z = y − y_{p−1}ϵ_{p−1} ∈ ∂H^{p−1}. We have

|y − c|^{2} = |z − w(α)|^{2} + (y_{p−1} − r_{α}/|q|^{2})^{2} ≤ (r_{α}/|q|^{2})^{2}.

This implies that

|y − w(α)|^{2} = |z − w(α)|^{2} + y_{p−1}^{2} ≤ 2y_{p−1}r_{α}/|q|^{2}

and

F_{y}w(α) = |q|^{2}|y − w(α)|^{2} ≤ 2y_{p−1}r_{α}.  □

From the inequality

2y_{p−1}|q||p − zq| ≤ |p − zq|^{2} + y_{p−1}^{2}|q|^{2}

we get that

Corollary 3. Let z be in ∂H^{p−1}. A chamber which intersects the line [z, ∞) has at least one parabolic vertex pq^{−1} satisfying

|pq^{−1} − z| ≤ \frac{1}{γ_{1}|q|^{2}} .  (5)

We have convergents satisfying (5). We now study a reciprocal result.

Proposition 1. For each parabolic vertex α ∈ C̃_{S}, let r_{α} > 0 be such that B_{α}(r_{α}) is in ∪_{S(α)}w(C̃_{S}). Let z be in ∂H^{p−1}. If pq^{−1} satisfies

|pq^{−1} − z| ≤ \frac{1}{γ_{2}|q|^{2}} ,  (6)

where γ_{2} = 1/ inf{r_{α}}, then pq^{−1} is a convergent.

Proof. Suppose that pq^{−1} = w(α) satisfies (6). The line [z, ∞) intersects the ball w(B_{α}(r_{α})). According to the definition of the B_{α}(r_{α}), there is f ∈ W_{S(α)} such that [z, ∞) intersects the chamber wf(C̃_{S}) with parabolic vertex pq^{−1}.

4. Reduced gallery and covering of a geodesic

We adopt the Tits vocabulary for buildings. The faces of the chambers w(C_{S}) are w(F_{i}) = w(C_{S} ∩ ∂A_{i}) and two chambers are adjacent if they have a common face. A gallery is a sequence (finite, or infinite indexed by ℕ or ℤ) of adjacent chambers. The length of a gallery is the number of its chambers. The distance d(C, C′) between two chambers C and C′ is the smallest length of a gallery with extremities C and C′ minus 1. It is clear that d(w_{1}(C_{S}), w_{2}(C_{S})) = l_{S}(w_{1}^{−1}w_{2}) for all w_{1} and w_{2}. We shall say that a gallery (C_{i}) is reduced if d(C_{i}, C_{j}) = |i − j| for all i and j.

We shall see that distances d and ρ are closely related by introducing galleries along geodesics.

We work with the Klein model. Let x_{1} and x_{2} be different in S^{p−2}. We denote by S_{succ}(x_{1}, x_{2}) the set of reflections s_{i} ∈ S satisfying

• x_{1} ∈ A_{i} and x_{2} ∈ s_{i}(A_{i})

• [x_{1}, x_{2}] is not in ∂A_{i}

• [x_{1}, x_{2}] ∩ F_{i} ≠ ∅ A necessary and sufficient condition for S_{succ}(x_{1}, x_{2}) ≠ ∅ is [x_{1}, x_{2}] ∩ C_{S} ≠ ∅. We define S_{pred}(x_{1}, x_{2}) := S_{succ}(x_{2}, x_{1}).

For different x_{1} and x_{2} in S^{p−2}, we denote by Arc(x_{1}, x_{2}) the set of w^{−1}[x_{1}, x_{2}] which intersect C_{S}. We introduce the oriented graph Γ(x_{1}, x_{2}) whose set of vertices is Arc(x_{1}, x_{2}) and edges (a, s, s(a)) for s ∈ S_{succ}(a), or equivalently (s(a), s, a) for s ∈ S_{pred}(a). The graph is simple and without loops.

Remark 1. The graph Γ(x_{2}, x_{1}) can be viewed as the reverse of Γ(x_{1}, x_{2}) by replacing the vertices by their opposites.

Remark 2. One can show that there is a one-to-one correspondence between finite graphs and closed geodesics in H^{n−1}/W_{S}.

Let σ = (σ_{0}, ⋯, σ_{N−1}) be a finite path in Γ(x_{1}, x_{2}) where σ_{n} = (a_{n}, t_{n+1}, a_{n+1}). To this path, we associate the gallery

G(σ) := (C_{S}, t_{1}(C_{S}), t_{1}t_{2}(C_{S}), …, t_{1} ⋯ t_{N}(C_{S})).

From (2), we deduce that the gallery G(σ) is reduced. In others words, l_{S}(w) is the number of hyperplanes between C_{S} and w(C_{S}).

From now on, we suppose that ∞ is a parabolic vertex of the fundamental chamber C̃_{S}. Let z be in ∂H^{p−1} and σ be an infinite path in Γ(v_{+}, Φ^{−1}(z)) indexed by ℕ with starting vertex a_{0} = [v_{+}, Φ^{−1}(z)]. Setting w_{0}(σ) = Id and w_{n}(σ) = t_{1} ⋯ t_{n} for n > 0, we obtain an infinite reduced gallery in the Poincaré half-space which covers the geodesic (z, ∞). We say that (w_{n}(σ)) in W_{S}^{ℕ} is an H-sequence associated to z (H for Hermite) or that we get (w_{n}(σ)) from the Hermite’s algorithm.

We need the following result Lemma 1. Let α be a parabolic vertex of C̃_{S} and t ∈ ℝ. The set

{|q| | pq^{−1} ∈ W_{S}(α), |q| ≤ t}

is finite.

Proof. Let B_{α}(r_{α}) be in the interior of ∪w(C̃_{S}) where w runs over W_{S(α)}. To each w ∈ W_{S}, we associate the ball w(B_{α}(r_{α})) = B_{pq^{−1}}(r_{α}/|q|^{2}). These balls are disjoint. Using the affine Euclidean subgroup W_{S(∞)}, we can suppose that pq^{−1} is in the bounded fundamental domain C̃_{S(∞)} ∩ ∂H^{p−1} for the action of W_{S(∞)} on ∂H^{p−1}. The set {|q| | pq^{−1} ∈ W_{S}(α), |q| ≤ t} is finite because only a finite number of disjoint balls with radius ≥ r_{α}/t^{2} can intersect the chimney C̃_{S(∞)}.

Now we are in position to prove

Theorem 2. Let z ∈ ∂H^{p−1} and (w_{n}) be a H-sequence associated to z. Let α be a parabolic vertex of C̃_{S}, and p_{n}q_{n}^{−1} := w_{n}(α). We have

(i) p_{n}q_{n}^{−1} is a convergent for z, (ii) the sequence n ↦ |p_{n} − zq_{n}| decreases, (iii) the sequence n ↦ |q_{n}| increases, and is bounded if and only if z is in W_{S}(α).

Proof. (i) See Corollary 1. (ii) Let u_{n} be the vector of t_{n}. From (4)

|p_{n+1} − zq_{n+1}|^{2} − |p_{n} − zq_{n}|^{2} = F_{z}w_{n+1}(α) − F_{z}w_{n}(α)

= −2\frac{B(Φ^{−1}(α), u_{n+1})B(w_{n}(u_{n+1}), Φ^{−1}(z))}{B(v_{+}, Φ^{−1}(z))} ≤ 0.

(iii) We suppose that |q_{n}| is bounded. Lemma 1 implies that there is an integer N such that |q_{n}| = |q_{N}| for n ≥ N. From (i) and (ii), we get that |p_{n} − zq_{n}| = |p_{N} − zq_{N}| for n ≥ N (see Corollary 1). Hence there is only a finite number of p_{n}q_{n}^{−1} (see proof of Lemma 1). One of these, say x := p_{N}q_{N}^{−1}, belongs to an infinite number of chambers in the gallery. This means that there is an infinite number of hyperplanes of reflection in W_{S(x)} which separate ∞ and z. We deduce that x = z.

We suppose that z = pq^{−1}. We know that |q_{n}| ≤ |q| from (i).

It is clear that lim_{n→∞} p_{n}q_{n}^{−1} = z, and the term “convergent” is now fully justified.

5. Approximation constants

For z in ∂H^{p−1}, we define the approximation constant γ(z) by

γ(z) = lim sup(|q||p − zq|)^{−1},

the limsup being taken over W_{S}(∞), and γ(z) = ∞ for z ∈ W_{S}(∞). We get a geometric interpretation of γ(z) by noting that (|q||p−zq|)^{−1} is the diameter of the half-circle w^{−1}(∞, z) in H^{p−1} where pq^{−1} = w(∞).

We say that z_{1} and z_{2} in ∂H^{p−1} are equivalent if there is w ∈ W_{S} such that z_{2} = w(z_{1}).

Proposition 2. If z_{1} and z_{2} are equivalent then γ(z_{1}) = γ(z_{2}).

Proof. We suppose that z_{1} and z_{2} are not equivalent to ∞. Let w ∈ W_{S} be such that z_{2} = w(z_{1}). Let (w_{n}) be a sequence in W_{S}^{ℕ} such that

γ(z_{2}) = lim_{n} |w_{n}^{−1}(z_{2}) − w_{n}^{−1}(∞)|.

We have the inequality

|w_{n}^{−1}(z_{2}) − w_{n}^{−1}(∞)| ≤ |w_{n}^{−1}w(z_{1}) − w_{n}^{−1}w(∞)| + |w_{n}^{−1}w(∞) − w_{n}^{−1}(∞)|.  (7)

We can write

|w_{n}^{−1}w(∞) − w_{n}^{−1}(∞)| = (|q_{n}|^{2}|w_{n}(∞) − w(∞)|)^{−1}

where p_{n}q_{n}^{−1} = w_{n}(∞). From the fact that lim_{n} |q_{n}| = ∞ and lim_{n} w_{n}(∞) = z_{2} ≠ w(∞), we get lim_{n} |w_{n}^{−1}w(∞) − w_{n}^{−1}(∞)| = 0. Hence from (7), we obtain γ(z_{1}) ≥ γ(z_{2}). By symmetry, we have γ(z_{1}) = γ(z_{2}).

From now on, we suppose that ∞ is the only parabolic vertex of the fundamental chamber.

Proposition 3. Let γ_{3} be the smallest diameter of edges of the fundamental chamber. Let z be in ∂H^{p−1} and (w_{n}) be a H-sequence associated to z. Setting w_{n}(∞) = p_{n}q_{n}^{−1}, we have

γ_{3} ≤ lim sup(|q_{n}||p_{n} − zq_{n}|)^{−1}.

Proof. We suppose that there is an integer N such that |w_{n}^{−1}(∞) − w_{n}^{−1}(z)| < γ_{3} for n ≥ N. There is a one-to-one correspondence between the vertices of C̃_{S} not in B_{∞}(1/γ_{3}) and the connected components of C̃_{S} \(C̃_{S} ∩B_{∞}(1/γ_{3})). Let α ∈ H^{p−1} be the vertex of the connected component encountered by w_{N}^{−1}(∞, z). We prove by induction on n that the reflections t_{n} used for the construction of the reduced gallery leave α invariant. But W_{S(α)} is a finite group. This gives the contradiction.

Now we know that γ_{3} ≤ γ(z) for all z. Hence γ_{3} is a lower bound of the Hurwitz constant C(W_{S}) = inf{γ(z), z ∈ ∂H^{p−1}} (see [11] in the general case of the geometrically finite discrete Möbius groups).

By noting that for any orbit W_{S}(y), y ∈ H^{p−1}, a point of the largest height in the orbit belongs to the fundamental domain C̃_{S}, we get immediately (see [12]):

Theorem 3. Let z be in ∂H^{p−1} and (w_{n}) be a H-sequence associated to z. We have

γ(z) = lim sup_{n} |w_{n}^{−1}(∞) − w_{n}^{−1}(z)|.

6. The main result about equivalence

We recall that, by assumption, ∞ is the only parabolic vertex of the fundamental chamber. We give here a necessary and sufficient condition for equivalence of two points by using the corresponding H-sequences.

Theorem 4. Let z_{1} and z_{2} in ∂H^{p−1} be not equivalent to ∞. Let (u_{n}) and (v_{n}) be the corresponding H-sequences. Then z_{2} = w(z_{1}) with w ∈ W_{S} if and only if the set {v_{n}^{−1}wu_{n}} is finite.

Proof. First, we suppose that z_{2} = w(z_{1}) with w ∈ W_{S}. Let p_{n}q_{n}^{−1} := u_{n}(∞). We have

|u_{n}^{−1}(∞) − u_{n}^{−1}w^{−1}(∞)| = (|q_{n}|^{2}|u_{n}(∞) − w^{−1}(∞)|)^{−1}.

We know that lim_{n} |q_{n}| = ∞ and lim_{n} |u_{n}(∞) − w^{−1}(∞)| = |z_{1} − w^{−1}(∞)| ≠ 0 because z_{1} is not in W_{S}(∞). This proves that lim_{n} |u_{n}^{−1}(∞) − u_{n}^{−1}w^{−1}(∞)| = 0. We deduce that for a sufficiently large n, there is a chamber w_{2}(C̃_{S}), with w_{2} in the finite set W_{2} = {w|w(C̃_{S}) ∩ C̃_{S} ∩ H^{p−1} ≠ ∅}, such that w_{2}(C̃_{S}) ∩ u_{n}^{−1}w^{−1}[∞, z_{2}] ≠ ∅. There is an integer p such that wu_{n}w_{2}(C̃_{S}) ∩ v_{p}(C̃_{S}) ≠ ∅. We deduce the existence of w′_{2} ∈ W_{2} such that wu_{n}w_{2} = v_{p}w′_{2}. It follows that

|n − p| ≤ l_{S}(w) + l_{S}(w_{2}) + l_{S}(w′_{2})

and

l_{S}(v_{n}^{−1}wu_{n}) ≤ |n − p| + l_{S}(w_{2}) + l_{S}(w′_{2}) ≤ l_{S}(w) + 4|W_{2}| < ∞.

We suppose that {v_{n}^{−1}wu_{n}} is finite. To simplify, we work with homographies and we set

w = ( \begin{matrix}a & b \\ c & d \end{matrix} ) , u_{n} = ( \begin{matrix}p_{n} & ∗ \\ q_{n} & ∗ \end{matrix} ) , v_{n} = ( \begin{matrix}P_{n} & ∗ \\ Q_{n} & ∗ \end{matrix} ) , v_{n}^{−1}wu_{n} = ( \begin{matrix}∗ & ∗ \\ γ_{n} & ∗ \end{matrix} ) .

A calculation shows that

|γ_{n}| = |Q_{n}||q_{n}||w(p_{n}q_{n}^{−1}) − P_{n}Q_{n}^{−1}||cp_{n}q_{n}^{−1} + d|.

We have lim_{n} |Q_{n}| = lim_{n} |q_{n}| = ∞ and lim_{n} p_{n}q_{n}^{−1} = z_{1} ≠ −c^{−1}d = w^{−1}(∞) because z_{1} is not in W_{S}(∞). By hypothesis, the set {|γ_{n}|} is finite. It follows that lim_{n} |w(p_{n}q_{n}^{−1})−P_{n}Q_{n}^{−1}| = 0. This proves that z_{2} = w(z_{1}).

7. An example

One can find examples in [7] (modular group, Hecke groups, Picard group...). Here we exhibit the most interesting. Following an idea of Prof. Y. Hellegouarch, we consider the hyperbolic reflection group W_{S} with Coxeter scheme

[figure]

This group is a discrete subgroup of Möb(H^{5}) acting on the Poincaré half-space of ℝ^{5}. The affine Weyl group W_{S(∞)} is of type D̃_{4}. The Clifford algebra Cl_{4} is generated by i, j, k and l such that i^{2} = j^{2} = k^{2} = l^{2} = −1 and ji = −ij … One can identify the reflections of S with

\begin{matrix}s_{1} = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )′ : x ↦ 1/x′, & s_{2} = ( \begin{matrix}−1 & 1 \\ 0 & 1 \end{matrix} )′ : x ↦ −x′ + 1, \\ s_{3} = ( \begin{matrix}−ξ′ & 0 \\ 0 & ξ \end{matrix} )′ : x ↦ −ξ′x′ξ′, & s_{4} = ( \begin{matrix}i & 0 \\ 0 & i \end{matrix} )′ : x ↦ −ix′i, \\ s_{5} = ( \begin{matrix}j & 0 \\ 0 & j \end{matrix} )′ : x ↦ −jx′j, & s_{6} = ( \begin{matrix}k & 0 \\ 0 & k \end{matrix} )′ : x ↦ −kx′k, \end{matrix}

where ξ = \frac{1}{2}(1 + i + j + k).

The entries of the Cliffordian matrices lie in ℤ[i, j, ξ]. We denote by Cl_{3} the sub-algebra generated by i, j and k, and by ℍ the sub-algebra generated by i and j. Considering the morphism Ψ : Cl_{3} → ℍ defined by Ψ(i) = i, Ψ(j) = j and Ψ(k) = ij, one can show that it is possible to work with the order 𝔒 = ℤ[i, i, \frac{1}{2}(1 + i + j + ij)] of the Hurwitz integers. If z is in Cg_{3} then |Ψ(z)| = |z|. Moreover Ψ establishes a one-to-one correspondence between the lattice ℤ[i, i, ξ] ∩ ℝ^{4} and 𝔒, and between the set of parabolic points and ℚ(i, j) ∪ {∞}. So the results of the previous sections can be stated as follows (γ_{1} = \sqrt{2} and γ_{2} = 2):

Theorem 5. Let z be in ℍ.

(i) For all α > 0, we can find p and q in 𝔒 such that |p − zq|^{2} + α^{2}|q|^{2} ≤ \sqrt{2}α. (ii) If p and q in 𝔒 satisfy |z − pq^{−1}| ≤ 1/2|q|^{2} then pq^{−1} is a convergent for z.

A lower bound for the Hurwitz constant C(W_{S}) is γ_{3} = \sqrt{5/2}. A. Schmidt showed in [9] that this bound is reached. We can find in [11] another proof which uses extremal geodesics. The author has developed in [7] a complete study of pseudo-periodic H-sequences which gives a third proof by showing that

γ (\frac{1}{2} + \frac{1 − \sqrt{5}}{4}i + \frac{1 + \sqrt{5}}{4}j) = \sqrt{\frac{5}{2}}.

The main idea is that, for a loxodromic fixed point, we can work with the corresponding closed geodesic.

References

[1] Ahlfors, L: Möbius transformations in ℝ^{n} expressed through 2 × 2 matrices of Clifford numbers. Complex Variables 5 (1986), 215-224.

[2] Beardon, A. F.:The Geometry of Discrete Groups. Graduate Texts in Math. 91, Springer-Verlag, New York 1983.

[3] Bourbaki, N.: Groupes et algèbres de Lie. Ch. 4–6. Hermann, Paris 1968.

[4] Ford, L. R.: Rational approximations to irrational complex numbers. Trans. Amer. Math. Soc. 19 (1918), 1–42.

[5] Hermite, C.: Sur la théorie des formes quadratiques. Crelle’s Journal f. d. Math. 47 [6] Humbert, G.: Sur la méthode d’approximation d’Hermite. Journal de Maths 2 (1916),

79–103.

[7] Meignen, P.: Groupes de Coxeter et approximation diophantienne. Thèse de l’Université de Caen, France, 1995.

[8] Meignen, P.: Orthochronous Lorentz Group from the Cliffordian point of view. Adv. Appl. Clifford Algebras 6 (2) (1996), 233–248.

[9] Schmidt, A. L.: Farey simplices in the space of quaternions. Math. Scand. 24 (1969), 31–65.

[10] Vinberg, E. B.: Geometry II. Encyclopedia Math. Sci. 29, Springer-Verlag, Berlin Heidelberg 1993.

[11] Vulakh, L. Ya.: Diophantine approximation in ℝ^{n}. Trans. Amer. Math. Soc. 347 (1995), 573–585.

[12] Vulakh, L. Ya.: Farey polytopes and continued fractions associated with discrete hyperbolic groups. Trans. Amer. Math. Soc., to appear.

Received January 12, 1998