On continuity of rotation numbers associated to geodesic laminations

H. K. Farahat Department of Mathematics

2500 University Drive Calgary T2N 1N4 Canada hfarahat@math.ucalgary.ca

I. V. Nikolaev Department of Mathematics 2500 University Drive Calgary T2N 1N4 Canada nikolaev@math.ucalgary.ca

July 30, 2003

Abstract

Geodesic lamination on the Riemann surface is an analogy of a family of parallel lines in the plane. There exists a notion of “slope” of this family on any Riemann surface ([6]). In this note we prove that the “slope function” is in fact a continuous function of the lamination. It is remarkable, that no geometric argument is required to establish this fact. We use exclusively postulates of continued fraction theory and arithmetic group theory.

Key words and phrases: Geodesic lamination, arithmetic group, continued fraction

AMS (MOS) Subj. Class.: 11F06, 20H10, 57M50

Introduction

Geodesic lines on the Riemann surfaces are critically important in the understanding of algebra and geometry of such surfaces. They have been extensively studied in the past by Hadamard ([3]), Artin ([1]), Morse ([4]) and others. Topologically, geodesic lines on the Riemann surface S can be subdivided into the three types: (i) immersed lines, i.e. lines with self-intersections on S; (ii) embedded periodic lines and (iii) embedded non-periodic lines on S. In this article we deal with the geodesic lines of types (ii) and (iii) and their topological closures on S, known as geodesic lamination. Let us remind the reader that geodesic laminations constitute a core of Thurston’s theory of 3-dimensional manifolds ([7]).

In ([6]) we attached to given geodesic lamination Λ certain real quantity α, called rotation number. Number α is a numerical invariant of lamination Λ which measures the average “slope” of geodesics relatively a fixed generator of the fundamental group of S. To this end, the rotation numbers can be viewed as a generalization of the Poincaré rotation numbers to the case S ≠ T^{2}.

For given geodesic lamination Λ its rotation number is a complete topological invariant of Λ. Namely, two geodesic laminations Λ, Λ′ on surface S are topologically equivalent if and only if their rotation numbers α, α′ are modular equivalent ([6]). This fact has various important applications in the theory of C^{∗}-algebras (classification of the AF C^{∗}-algebras), algebraic geometry (moduli problem of Riemann surfaces) and geometric topology (classification of 3-dimensional manifolds).

If S is an arithmetic Riemann surface^{1} , then the rotation number, α, has especially nice presentation. Namely, α unfolds into the regular continued fraction

α = a_{0} + \frac{1}{a_{1} + \frac{1}{a_{2} + …}} = [a_{0}, a_{1}, a_{2}, …],  (1)

where a_{i} = a_{i}(Γ_{n}) are positive integers defined by the principal congruence group Γ_{n}. In fact, this is exactly the way rotation numbers were introduced in ([6]).

The aim of present note is to study the range of the “rotation function” defined by equation (1). Namely, for fixed principal congruence group Γ_{n}, what values of α will appear in formula (1)? The answer is contained in the following statement, which to the best of our knowledge has not been studied in the literature.

1We call S arithmetic if S = ℍ/Γ_{n}, where ℍ is the Lobachevsky plane and Γ_{n} the principal congruence group of order n, see Section 1 for the definition.

Theorem 1 (Surjectivity of rotation function) Let n ≥ 1 be an integer number. Suppose that S is an arithmetic Riemann surface S = ℍ/Γ_{n}, where Γ_{n} is the principal congruence (sub-)group of step n of the modular group SL_{2}(ℤ). Then for any irrational number α ∈ ℝ, there exists a geodesic lamination Λ on S whose rotation number is equal to α. In other words, geodesic laminations on S can have any (irrational) slope on Riemann surface S.

1 Notation

In this section we briefly review notation and definitions to be used in this note. We refer the reader to ([4]), ([5]) and ([6]) for the details.

1.1 Geodesics

We identify Riemann surfaces of genus g ≥ 2 with the compact surfaces of constant negative curvature −1. Any such a surface will be denoted by S. By a geodesic on S we understand the maximal arc on S consisting of the locally shortest sub-arcs (in the given metric on S). Topologically, each geodesic is the image of the open real interval I under a continuous mapping I → S.

Using symbolic dynamics Artin, Morse and Myrberg established existence of the following types of geodesic lines on the surfaces of constant negative curvature.

Lemma 1 (Topological classification of geodesics) Let p ∈ S be a point with an attached unit vector t ∈ S^{1} on the surface S. Then :

(a) for almost all points t ∈ S^{1} the geodesic line through p in direction t is an immersion I → S, i.e. finite or infinite curve with self-intersections;

(b) the complementary set 𝔎 ⊂ S^{1} has the cardinality of continuum and geodesic lines through p in direction t ∈ 𝔎 are embedded curves of one of the three types:

(i) periodic (ii) spiral (iii) non-periodic filling-up the entire S (Cantor-type curve).

Proof. See Artin ([1]) and Myrberg ([5]) for the original proof. □

In the sequel, we mainly work with the geodesics of type (iiib) and following Birkhoff ([2]) and Morse ([4]) call such lines recurrent geodesics. As the reader will see, recurrent geodesics are a replacement for the “straight lines” on the Riemann surfaces S ≠ T^{2}. The topological closure of a recurrent geodesic on S contains continuum of distinct recurrent geodesics on S (having the same closure), and such a collection of geodesics is called (minimal) geodesic lamination. We shall discuss laminations in Section 1.4.

1.2 Arithmetic Riemann surfaces

Denote by Γ the modular group P SL_{2}(ℤ). For every n = 1, 2, 3, …, consider its normal subgroup Γ_{n} ⊂ Γ (of finite index) consisting of matrices of the form:

( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) Mod n = ( \begin{matrix}1 + np & nq \\ nr & 1 + ns \end{matrix} ) ,  (2)

where p, q, r, s ∈ ℤ. The subgroup Γ_{n} is called a principal congruence group of step n.

Let ℍ = {z = x+iy ∈ ℂ|y > 0} be the Lobachevsky (half-) plane endowed with the hyperbolic metric ds = |dz|/y. The principal congruence group Γ_{n} acts discretely on ℍ by the linear-fractional transformations z ↦ \frac{az+b}{cz+d}, where

( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{n}. The orbit space S = ℍ/Γ_{n} is an orbifold with a finite number

of cusps corresponding to the (finite) stabilizers of group Γ_{n}. It follows from the construction that S is a Riemann surface, which we call an arithmetic Riemann surface to underline its connection with the arithmetic subgroups of the modular group. The genus of S can be evaluated according to the classical formula:

g = 1 + \frac{1}{24}τn^{2}(n − 6) ∏_{p|n} (1 − \frac{1}{p^{2}} ) ,  τ = \{ \begin{matrix}2, if n = 2, \\ 1, if n > 2 \end{matrix} .  (3)

1.3 Rotation numbers

Assume that S is an arithmetic Riemann surface of genus g ≥ 2. We wish to associate a real number α to every recurrent geodesic γ on S. The construction is based on the continued fraction, which arises from an infinite sequence {g_{i} | g_{i} ∈ Γ_{n}, i = 1, …, ∞} and is attached “almost uniquely” to γ.

Namely, take an arbitrary fundamental region (orbit space) Ω of fixed principal congruence group Γ_{n} (Fig. 1). Since γ is recurrent, it intersects Ω infinitely often, and let us write down the corresponding arcs of intersection as {I_{0}, I_{1}, …} in the order they cut Ω. Since each I_{k} is the arc of the geodesic half-circle on ℍ, they can be moved to each other by transformations from Γ_{n}. Put then g_{k} ∈ Γ_{n} be such that I_{k} = g_{k}(I_{k−1}). It can be shown ^{2} that g_{k} ∈ Γ_{n}^{+}, where Γ_{n}^{+} is the principal congruence semi-group consisting of 2 × 2 matrices with positive integer entries. Finally, suppose that g_{k} =

( \begin{matrix}a_{0}^{(k)} & 1 \\ 1 & 0 \end{matrix} ) … ( \begin{matrix}a_{n}^{(k)} & 1 \\ 1 & 0 \end{matrix} ) is the Minkowski (prime) decomposition of matrix

g_{k} ∈ Γ_{n}^{+}. Then the regular continued fraction

a_{0}^{(0)} + \frac{1}{a_{1}^{(0)} + \frac{1}{a_{2}^{(0)} + …}} ,  (4)

converges to an irrational number α called a rotation number associated to the recurrent geodesic γ.

1.4 Topological equivalence of geodesic laminations

The closure of any recurrent geodesic on S contains infinitely many (a continuum) of other recurrent geodesics. The disjoint union of such geodesics gives us a geodesic lamination, Λ, on surface S. Λ can be thought of as collection of “parallel” geodesic lines, i.e lines with the same asymptotic behaviour. It can be easily seen that the rotation number associated with any geodesic in Λ will be the same.

On the other hand, there exists infinitely many topologically distinct geodesic laminations on given surface S. By the topological equivalence of two laminations Λ, Λ′ one means a homeomorphism h : S → S which preserves the leaves of the laminations and sends leaves of Λ to the leaves of Λ′. It turns out that the topological equivalence of laminations is “controlled” be their rotation numbers in the following sense.

2This fact is an implication of Selberg’s theory of the length spectra of Riemann surfaces. Indeed, the iteration process of recurrent geodesic γ by segments I_{k} would give us an infinite sequence of periodic geodesics {p_{k} | k = 1, …, ∞}, which converges to γ. By Selberg’s formula, the length of geodesic p_{k} is 2 cosh(\frac{1}{2}l(p_{k})) = |tr(G_{k})|, where G_{k} = ∏_{j=1}^{k} g_{j}. Since the lengths of periodic geodesics p_{k} grow, so do traces tr (G_{k}).

This can happen if and only if all the entries of matrices g_{k} are non-negative integers. In other words, “prime decomposition” of the recurrent geodesic into a continued fraction is morally equivalent to decomposition of Riemann surfaces into their “length spectra”.

Lemma 2 (Topological equivalence of geodesic laminations) Two geodesic laminations Λ, Λ′ are topologically equivalent if and only if their associated continued fractions coincide, except possibly in the finite number of terms. In other words, the associated rotation numbers α, α′ are modular equivalent:

α′ = \frac{aα + b}{cα + d},  a, b, c, d ∈ ℤ,  ad − bc = ±1.  (5)

Proof. See ([6]) for the proof. □

2 Proof of Theorem 1

Note that geometric interpretation of Theorem 1 is intuitively obvious and requires no formal proof. (Indeed, the slope of a geodesic lamination should be a continuous quantity, in the sense that “close” geodesic laminations must have infinitely close slopes.) We wish to give here an algebraic proof of this fact.

Our proof is based on the fact that Γ_{n} is the subgroup of finite index in the modular group Γ = P SL_{2}(ℤ). Or, in geometric terms, the arithmetic Riemann surface S_{n} = ℍ/Γ_{n} finitely covers the modular surface S = ℍ/Γ.

Let us consider the following semi-group:

Γ^{+} = \{ ∏_{i even} ( \begin{matrix}a_{i} & 1 \\ 1 & 0 \end{matrix} ) | a_{i} ∈ ℕ\} ,  (6)

where the product is taken over finite even number of factors. The following lemma (Lemma 3) shows that every non-negative integer matrix of determinant +1 belongs to Γ^{+} and vice versa.

Lemma 3 (Minkowski’s lemma) Let A = ( \begin{matrix}a & b \\ c & d \end{matrix} ) be an integral matrix with the non-negative entries and det A = 1. Then there exists a unique decomposition of A into the even number of “prime factors” ( \begin{matrix}a_{i} & 1 \\ 1 & 0 \end{matrix} ), where a_{i} ≥ 0. In other words,

( \begin{matrix}a & b \\ c & d \end{matrix} ) = \underbrace{( \begin{matrix}a_{1} & 1 \\ 1 & 0 \end{matrix} ) … ( \begin{matrix}a_{n} & 1 \\ 1 & 0 \end{matrix} )}_{even}  (7)

Proof. See Wall ([8]) □

Now we are able to establish the range of the “rotation function” for the modular surface. Namely, the following lemma is true.

Lemma 4 (Surjectivity of rotation function for modular surface) Let S = ℍ/Γ be the modular surface. Then for any irrational number α ∈ ℝ\ℚ, there exists a geodesic lamination Λ ⊂ S whose rotation number is equal to α.

Proof. Let z ↦ T z be linear fractional transformation T of the complex plane ℂ. It is known that under the action of infinite product ∏_{i=1}^{∞} T_{i}, point z = ∞ goes to the point z′ ∈ ℂ whose coordinates unfold into an infinite continued fraction, convergent or not, see ([8]), p. 13. In the important case T_{i} ∈ Γ^{+}, the action has especially nice form:

∏_{i=1}^{∞} T_{i}(∞) = ∏_{i=1}^{∞} T_{i}(0) = a_{0} + \frac{1}{a_{1} + \frac{1}{a_{2} + …}} ,  (8)

where a_{i} are entries of the prime factors of T_{i} (see Lemma 3). Continued fraction at the right of equation (8) always converges to a real (irrational) number, and therefore points z = 0 and z = ∞ go to an irrational point on the real line ℝ ⊂ ℂ. On the other hand, by the Lagrange theorem any irrational number unfolds uniquely into regular continued fraction (8). Lemma 4 follows. □

Remark 1 Geometrically, Theorem 1 follows from Lemma 4. Indeed, since given arithmetic Riemann surface S_{n} finitely covers modular surface S, so does any geodesic lamination Λ′ on S_{n}. Namely, Λ′ covers a geodesic lamination Λ ⊂ S, for which the surjectivity of rotation numbers is established by Lemma 4. It remains to “lift” the obtained family of rotation numbers to the surface S_{n}. However, we wish to give purely algebraic proof of Theorem 1.

To finish the proof of Theorem 1 we would need the following key lemma.

Lemma 5 Let G be a finite group. Then every infinite word on the elements of G is equivalent to the infinite word 1_{G}1_{G} … consisting of the repeating identity 1_{G} of group G.

Proof. Let w = g_{1}, g_{2}, … be an infinite word on the elements g_{i} of group G. Consider the “partial products” p_{1} = g_{1}, p_{2} = g_{1}g_{2}, ... p_{n} = g_{1}g_{2} … g_{n}, .... We proceed by induction.

Step 1. If p_{1} = p_{2}, then g_{2} = 1_{G}. In this case w = g_{1}1_{G}g_{3} … = 1_{G}g_{1}g_{3} … by commutativity of the identity element 1_{G}. We resume the procedure for the word w′ = g_{1}, g_{3}, . . ..

Step 2. If p_{1} ≠ p_{2}, then suppose p_{1} = p_{3}. In this case g_{2}g_{3} = 1_{G} and we get the word w = g_{1}1_{G}g_{4} … = 1_{G}g_{1}g_{4} … and repeat the procedure for the word w′ = g_{1}g_{4} . . ..

Step n. Since G is a finite group, there exists a number n ≤ |G| such that p_{1} = p_{n}. Then the algorithm eventually stops, and as it was shown earlier, one obtains the word w = 1_{G}g_{1}g_{n+1} . . ..

The induction is repeated, so that any infinite word w reduces to the form 1_{G}1_{G} … Lemma 5 follows. □

Let G = Γ/Γ_{n} be the factor group of the modular group by the principal congruence group of step n. It is known that G is finite group. Thus Γ splits into a finite union of cosets gΓ_{n}, with the well-defined group operation between the cosets. Note that Γ_{n} is a coset itself corresponding to the identity 1_{G} of group G.

Finally, take an irrational number α ∈ ℝ\ℚ. By Lemma 4,

α = ∏_{i=1}^{∞} T_{i}(∞) = ∏_{i=1}^{∞} T_{i}(0),  (9)

and one gets a correctly defined infinite sequence {T_{i} | i = 1, 2, … ∞} of elements T_{i} ∈ Γ. If we replace each T_{i} by the corresponding coset, we get a word {g_{i} | i = 1, 2, … ∞}, where g_{i} ∈ G are elements of finite group G. By Lemma 5 the latter word is equivalent to the infinite word 1_{G}1_{G} . . .. In other words, the product (9) can be rearranged to the product:

α = ∏_{i=1}^{∞} 𝔗_{i}(∞),  (10)

where all 𝔗_{i} are elements of the principal congruence group Γ_{n}. Theorem 1 is proven. □

References

[1] E. Artin, Ein mechanishes System mit quasiergodischen Bahnen, Abh. Math. Sem. Hamburg (1924), 170-175; in: Collected papers, 499-504, Springer Verlag.

[2] G. D. Birkhoff, Quelques théorèmes sur le mouvement des systèmes dynamiques, Bull. de la Soc. Math. de France 40 (1912), 305-323.

[3] J. Hadamard, Les surfaces à courbures opposées et leur lignes geodesiques, Journal de Mathématique pures et appliquées 4 (1898), 27-73.

[4] M. Morse, Recurrent geodesics on a surface of negative curvature, Trans. Amer. Math. Soc. 22 (1921), 84-100.

[5] P. J. Myrberg, Ein Approximationssatz für die Fuchsschen Gruppen, Acta Math. 57 (1931), 389-409.

[6] I. Nikolaev, Geometry of a crossed product, Proc. Amer. Math. Soc. 128 (2000), 1177-1183.

[7] W. P. Thurston, The Geometry and Topology of Three-Manifolds, MSRI 1997, electronic edition of 1980 Princeton Univ. notes, available at http://www.msri.org/gt3m/

[8] H. S. Wall, Analytic theory of continued fractions, Chelsea, N.Y., 1973