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On the homeomorphisms of the space of geodesic laminations on a hyperbolic surface

We prove that for any orientable connected surface of finite type which is not a a sphere with at most four punctures or a torus with at most two punctures, any homeomorphism of the space of geodesic laminations of this surface, equipped with the Thurston topology, is induced by a homeomorphism of the surface.


Introduction
In this paper S = S g,p is an orientable connected surface of finite type, of genus g ≥ 0 with p ≥ 0 punctures. We assume that S is not a sphere with at most four punctures or a torus with at most two punctures. Fixing a complete hyperbolic metric of finite area on S we consider the set GL(S) of geodesic laminations on S with compact support. As a set of compact subspaces of the metric space S, GL(S) is equipped with the Hausdorff metric which we denote by d H . We denote by T H the topology induced by d H , and we call it the Hausdorff topology. We consider on GL(S) a second topology T, referred to as the Thurston topology (see the definition in §2 below) and which is weaker than T H . Any homeomorphism h : S → S induces by push-forward a map h * : GL(S) → GL(S) which is a homeomorphism with respect to both topologies, T H and T. The main result of this paper is the following: Theorem 1.1. The natural homomorphism from the extended mapping class group of S to the group of homeomorphisms of GL(S) equipped with the Thurston topology, is an isomorphism.
In particular, if f : GL(S) → GL(S) is a homeomorphism with respect to the topology T, then there is a homeomorphism h : S → S such that h * = f. This shows that the space GL(S) equipped with the Thurston topology is definitely not topologically homogeneous. Furthermore, since its homeomorphism group is countable, this space does not contain any open set which is a manifold of any positive dimension, in contrast with measured lamination space which is topologically a sphere.
The analogous result for the Hausdorff topology is not true but it is conceivabe that for every isometry f of the metric space (GL(S), d H ), there is a homeomorphism h : S → S such that h * = f.
The result is in the spirit of several rigidity results that were obtained by various authors in the context of mapping class group actions on different spaces, and it is close more specially to the results in [11], [8] and [9] which concern actions by homeomorphisms. We note that although the statements of these and other rigidity theorems all look alike, all of them are interesting because each proof displays new features of the space on which the mapping class group acts, in different settings (combinatorial, topological, holomorphic, metric, etc.), and the arguments in each setting are often different.
The proof of Theorem 1.1 is given in §4. It involves the construction from the homeomorphism f an automorphism of the complex of curves of S. The main technical point is to show that f preserves the set of finite laminations. For the passage between properties of laminations and homeomorphisms of GL(S), the idea is to translate inclusions Λ 1 ⊂ Λ 2 between geodesic laminations into set-theoretical properties between open sets of GL(S).
We would like to thank Ken'ichi Ohshika who read a preliminary version of this paper and corrected several mistakes, and Yi Huang who pointed out a gap in a previous version.

Definitions and Preliminaries
On the surface S, we fix a complete hyperbolic metric of finite area. A geodesic lamination Λ ⊂ S is a compact non-empty subset which is the union of disjoint simple geodesics. We say that a geodesic lamination is maximal if it is not a proper sublamination of any other geodesic lamination. We say that a geodesic lamination is minimal if it does not contain any proper sublamination. Note that this settheoretic definition of a minimal lamination is not the same as the usual definition of a minimal laminations, since usually a lamination is said to be minimal if its leaves are dense in the support.
A geodesic lamination which is a finite union of geodesics is called a finite lamination. Otherwise, it is said to be infinite.
The following two subsets of GL(S) will play special roles in the sequel: • FGL(S) is the subset of finite laminations of GL(S) . An element of FGL(S) is made out of a finite union of disjoint simple closed geodesics {γ i } together with a finite number of infinite geodesics, each spiraling from each end around one geodesic in {γ i }.
• CGL(S) is the set of geodesic laminations whose leaves are simple closed geodesics.
For any subset X of S and for any ǫ > 0, we set N ε (X) = {x ∈ S : ∃y ∈ X with d(x, y) < ε}.
The following definition is classical: Definition 2.1. Let X and X ′ be two compact subsets of S. The Hausdorff distance between X and X ′ is the quantity It is easy to see that the function d H is a distance function on the set of compact subsets of S. Such a definition was made by F. Hausdorff for the set of compact subsets of R n , and it was used by H. Busemann [1] for the set of compact sets of a general metric space.
We shall mostly use the notions N ε (X) and d H (X, Y ) for elements X, Y ⊂ S which are geodesic laminations on S. We also denote by d H the restriction of the Hausdorff metric to GL(S).
We also use the following notation: For any Λ ∈ GL(S) and for any ε > 0, The topology induced by d H on the set of subsets of S as well as its restiction to GL(S), which we shall denote by T H , are called the Hausdorff topology. For the topology T H any set V ε (Λ) is open. Moreover, it is easy to see that the collection of sets V ε (Λ) for all ε > 0 and Λ ∈ GL(S), constitute a basis for T H .
We now equip the set GL(S) with a second topology.
We let T be the topology on GL(S) with subbasis the sets U V , where V varies over the set of open subsets of S.
Following the terminology of ( [2], Def. I.4.1.10), we call the topology T on GL(S) the Thurston topology. The original reference for the topology T is (Thurston [12], Section 8.10), where T is referred to as the geometric topology. Clearly the topology T does not satisfy the first axiom of separation. Indeed, take a geodesic lamination Λ that contains a strict sublamination Λ 1 Λ 2 ; then every open set for T containing Λ 1 contains Λ. In particular the topology T is not Hausdorff, unlike the topology T H , which is induced by a metric.

Proof. It suffices to prove that for each open subset
For any x ∈ Λ ∩ V , there exists an open ball B(x, ε Λ ) in S of center x and radius ε Λ such that B(x, ε Λ ) ⊂ V . We now prove the following: First, it is obvious that Λ ∈ V εΛ (Λ). To prove the inclusion V εΛ (Λ) ⊂ U V we note that if Λ ′ ∈ V εΛ (Λ) then Λ ⊂ N εΛ (Λ ′ ) and hence x ∈ N εΛ (Λ ′ ). Therefore we can find a point y in Λ ′ such that d(x, y) < ε Λ and hence y ∈ Λ ′ ∩ B(x, ε Λ ) = ∅. Since B(x, ε Λ ) ⊂ V, this implies that Λ ′ ∩ V = ∅. Therefore Λ ′ ∈ U V and the inclusion V εΛ (Λ) ⊂ U V is proven. Now from (1) we deduce immediately that The metric space (GL(S), d H ) is compact. This is a general result on the Hausdorff metric on the set B(X) of compact subsets of a metric space X, and GL(S) is a closed subset of B(S) (cf. [3],Theorem 3.4 for this special case). Therefore T H is a compact topology. Since T ⊂ T H , it follows that T is also a compact topology.
In the next theorem we summarize basic properties of minimal geodesic laminations and we also give a description of the structure of maximal geodesic laminations. The properties are all well known from Thurston's theory, and we give references for the convenience of the reader. (II) (Corollary A.2.4, p. 143 in [10] or Lemmata 4.2 and 4.3 in [3]) Let Λ be a minimal geodesic lamination with infinitely many leaves. Then every leaf of Λ is dense in Λ. Furthermore, Λ contains a finite number of leaves which are isolated from one side. These leaves appear as boundary geodesics of some C(U ), where U is a component of S − Λ; they will be referred to as boundary leaves of Λ.
(III) (Theorem I.4.2.8, p. 83 in [2]) Let Λ be an arbitrary geodesic lamination of S. Then Λ consists of the disjoint union of a finite number of minimal sublaminations of Λ together with a finite set of additional geodesics each end of which spirals onto a minimal lamination. Each of the additional geodesics is isolated, i.e. it is contained in an open subset of S which is disjoint from the rest of the lamination.
In Item (III) above, the fact that an end of a geodesic spirals on a minimal sublamination Λ ′ of Λ means that the set of accumulation points of this end on the surface is Λ ′ .
The following lemma and proposition will also be used in the next section.
Lemma 2.5. Let Λ be an infinite minimal geodesic lamination of S. Then Λ has at least two boundary leaves.
Proof. The lift of Λ to the universal cover of S is a geodesic laminationΛ of the hyperbolic plane. Each component of the complement ofΛ is an ideal polygon. The images in S of two boundary leaves of such a polygon give the desired boundary leaves of Λ.
Proposition 2.6 (see [2], I 4.2.14 p. 81). The finite laminations are dense in the space of geodesic lamnitations equipped with the Hausdorff topology. Hence the same holds for the Thurston topology.

On the action of a homeomorphism of GL(S)
We denote by O(S) the set of open subsets of S and we fix an element Λ of GL(S). We consider the sets We have the following.
Conversely, assume Λ 2 ∈ U(Λ 1 ). If Λ 1 is not a subset of Λ 2 , then there exists Proof. It suffices to show that To prove the assertion, we start by Therefore, we have From the above lemmata, we obtain the following: Lemma 3.5. Let f be a homeomorphism of GL(S) with respect to the Thurston topology. Then f sends a maximal (respectively minimal) geodesic lamination of S to a maximal (respectively minimal) geodesic lamination of S.
Proof. Let Λ be a maximal geodesic lamination of S. If f (Λ) is not maximal then there is a maximal geodesic lamination Θ such that f (Λ) Θ. Let Λ ′ = f −1 (Θ). Then, by Corollary 3.4, we have Λ Λ ′ which contradicts the maximality of Λ.
From Theorem 2.4 (III), every Λ ∈ GL(S) has a finite number of sublaminations. Thus, we give the following definition.
We denote such a chain by C Λ . The integer n will be called the length of C Λ and will be denoted by l(C Λ ).
A chain of sublaminations C Λ will be called maximal if its length is maximal among all chains of sublaminations of Λ.
The length of a maximal chain of sublaminations C Λ of Λ depends only on Λ. Therefore the number l(C Λ ) will be referred to as the length of Λ and will be denoted by length(Λ). Lemma 3.7. Let f be a homeomorphism of GL(S) with respect to the Thurston topology and let Λ n Λ n−1 ...
Proof. The proof follows immediately from Corollary 3.4.
We call a generalized pair of pants a hyperbolic surface which is homeomorphic to a sphere with three holes, a hole being either a geodesic boundary component or a cusp. Now we can prove the following proposition. (1) f sends any maximal finite lamination which contains a collection of curves that decompose S into generalized pair of pants to a maximal finite lamination that has the same property.
(2) f sends any laminations whose leaves are all closed to a lamination having the same property. Furthermore if such a lamination Λ has k components then f (Λ) has also k components. A geodesic of type (ii) appears as a boundary leaf of a minimal infinite sublamination of Λ. Among the {λ i }, we may assume, without loss of generality, that λ 1 , ..., λ k are the isolated leaves of Λ, for 0 ≤ k ≤ n. From Theorem 2.4, it follows that if Λ is a finite maximal geodesic lamination then k = n and that if Λ is a maximal infinite geodesic lamination which is also minimal then k = 0.
We define a generating set for a geodesic lamination Λ to be a set A = {µ 1 , . . . , µ k } of leaves of Λ such that the union of the closures of the leaves that belong to A is the lamination Λ. It follows from Theorem 2.4 that every geodesic lamination on S has a finite generating set.
Our terminology is motivated by Claim 2 that follows now. Claim 2: The set A Λ = {λ 1 , .., λ n , c 1 , .., c m } is a generating set for Λ and any proper sublamination of Λ is the union of closures of leaves that belong to some proper subset of A Λ .

Proof of Claim 2. The claim follows immediately from the definition of A and from Theorem 2.4 (III).
It is well-known and easy to see, using an Euler characteristic count, that the maximum number of pairwise disjoint simple closed geodesics in S is equal to 3g − 3 + b and that these geodesics cut S into 2g − 2 + b hyperbolic generalized pairs of pants. It is also easy to see that the maximum number of open geodesics l i of S which decompose S into hyperbolic ideal triangles is equal to 6g − 6 + 3b.
Claim 3: If Λ is a maximal finite geodesic lamination that contains a generalized pair of pants decomposition P then a maximal chain of sublaminations of Λ has length 9g − 9 + 3b. Proof of Claim 3. The lamination Λ contains 3g − 3 + b simple closed geodesics, say c 1 , .., c 3g−3+b , which cut S into generalized pairs of pants, and additional open isolated geodesics, say λ 1 , ..., λ r , such that each λ i spirals about some c j . We may add to λ i open geodesics λ ′ k , which from one direction abut to a cusp and from the other direction spiral about some c j , such that all the geodesics λ i and λ ′ i decompose S into hyperbolic ideal triangles. Since the total number of λ i and λ ′ i is equal to 6g − 6 + 3b we deduce that the number r of λ i is equal to 6g − 6 + 2b. Therefore the set A Λ = {λ 1 , .., λ 6g−6+2b , c 1 , .., c 3g−3+b } is a generating set of Λ. From Claim 2, a chain of sublaminations Λ n Λ n−1 ... Λ 1 Λ 0 = Λ is maximal if and only if for each k = 1, .., n, Λ k − Λ k−1 is a single leaf belonging to A Λ . Such a sequence Λ i can be constructed as follows: From Λ 0 = Λ we first remove, one by one, all leaves λ i ; after removing all these leaves we continue removing, one by one, all leaves c j . Obviously n = 9g − 9 + 3b. Claim 4: Let Λ be a maximal finite geodesic lamination which does not contain a generalized pants decomposition. Then length(Λ) < 9g − 9 + 3b. Proof of Claim 4. Since Λ does not contain a generalized pants decomposition it follows that if c 1 , . . . , c k are the closed geodesics in Λ, then k is strictly smaller than 3g − 3 + b. Now as in the proof of Claim 3, we may prove that Λ contains additional open geodesics λ 1 , . . . , λ 6g−6+2b which spiral about the c j 's. Obviously, a maximal chain C Λ of sublaminations of Λ has length less than 9g − 9 + 3b. Claim 5: Let Λ be a maximal infinite geodesic lamination. Then length(Λ) < 9g − 9 + 3b.
Consider a maximal chain of sublaminations of Λ, say Λ k Λ k−1 ... Λ 1 Λ 0 = Λ. Let A Λ = {λ 1 , .., λ n , c 1 , .., c m } be a generating set of Λ, where each λ i is an open geodesic and each c i is a closed geodesic, and we assume that the generating set A Λ is minimal in the sense that no proper subset of A Λ is a generating set of Λ.
First, we prove that m < 3g−3+b. We know that m ≤ 3g−3+b. If m = 3g−3+b then the closed geodesics cut S into generalized pairs of pants. This implies that a minimal sublamination, say Λ ′ , of Λ with infinitely may leaves must be in the interior of a generalized pair of pants. But it is easy to see that such a sublamination Λ ′ does not exist. Therefore m < 3g − 3 + b.
Second, we prove that n ≤ 6g − 6 + 2b. By Theorem 2.4, the leaves λ 1 , .., λ n are either isolated open leaves or boundary leaves. Since Λ is maximal, for each component U of S − Λ the completion C(U ) is either a hyperbolic ideal triangle or a cusped hyperbolic monogon. The area of every such surface is equal to π, therefore, from the Gauss-Bonnet theorem the number of components U is equal to 4g − 4 + 2b. Now, each hyperbolic ideal triangle has three sides and each cusped hyperbolic monogon has one side. On the other hand the number of cusped hyperbolic monogons is equal to b. Therefore the total number of sides of C(U ) is 3(4g − 4 + 2b) − 2b = 12g − 12 + 4b. Now if a leaf λ i is isolated it belongs exactly to two components C(U ). If a leaf λ i is not isolated then it belongs to a minimal sublamination of Λ, say K i . From Lemma 2.5, K i contains at least two boundary leaves and we take λ i to be one of them. Therefore, among all the sides of the given C(U ), at least two of them are boundary leaves of K i . This implies that n ≤ 12g−12+4b 2 = 6g − 6 + 2b. Assume that λ 1 , ..., λ r ∈ A Λ , 0 ≤ r < n are isolated leaves and λ r+1 , ..., λ n ∈ A Λ are isolated from one side. Every λ s with s > r belongs to a unique minimal infinite sublamination Λ ′ s of Λ. Now the maximal chain of sublaminations Λ k Λ k−1 ... Λ 1 Λ 0 = Λ of Λ is constructed as follows: We pass from Λ i−1 to Λ i by removing all the isolated open geodesics λ i ∈ A Λ with i ≤ r, then every minimal infinite sublamination Λ ′ t of Λ and every closed geodesic c j ∈ A Λ . Obviously the length of Λ is less than 9g − 9 + 3b. This finishes the proof of Claim 5. Now, from Lemma 3.7 the homeomorphism f preserves the length of a maximal lamination Λ and this finishes the proof of (1). Now we prove (2). First, we claim that if K is a geodesic lamination consisting of k closed geodesics then f (K) is a finite lamination. To prove this, we consider a maximal finite geodesic lamination Λ containing a generalized pair of pants decomposition P with K ⊂ P . By (1), f (Λ) is a maximal finite lamination and thus our claim follows. From this claim and Lemma 3.7, it follows that if K 1 = {c} is a geodesic lamination consisting of a single closed geodesic then the image of K 1 by f consists of a single closed geodesic. Now we have proved Statement (2) for n = 1 and we proceed by induction. We assume that the statement is true for all n ≤ k and we prove it for n = k + 1.
Assume that K consists of k + 1 closed geodesics. If our statement were not true for n = k + 1, then from our inductive assumption and Lemma 3.7, f (K) would consist of k closed geodesics, say c 1 , · · · , c k , plus one open geodesic. Then, any sublamination of K consisting of k geodesics would be sent to {c 1 , · · · , c k }. But this is impossible since f is a homeomorphism. This completes the induction and the proof of (2). Now we prove (3). Let K be a finite lamination. Obviously, a minimal sublamination of K consists of a single closed geodesic. Assume that f (K) is not finite. Then there is an infinite minimal sublamination Λ 0 of f (K). From Lemma 3.5, Λ 0 is the image of a minimal sublamination K 0 of K, via f , i.e. f (K 0 ) = Λ 0 . But from (2), f (K 0 ) consists of a single closed geodesic. Therefore we have a contradiction, which proves (3).

Proof of Theorem 1.1
We first prove that if f : GL(S) → GL(S) is a homeomorphism with respect to the topology T, then there is a homeomorphism h : S → S such that h * = f.
We need the following lemma.
Lemma 4.1. Assume that (Λ n ) is a sequence of laminations that converges to a lamination Λ with respect to the Hausdorff topology. Assume also that (Λ n ) converges to Λ ′ with respect to the Thurston topology. Then Λ ′ ⊂ Λ.
Proof. If Λ ′ is not contained in Λ then there exists a point x ∈ Λ ′ − Λ. Therefore there exists an open ball B(x, 2ε) in S of center x and radius 2ε such that B(x, 2ε)∩ Λ = ∅. We also have Λ ′ ∈ U B(x,ε) . Now, since Λ n → Λ with respect to the Hausdorff topology we deduce that there is n 0 ∈ N such that the following holds: On the other hand, Λ n → Λ ′ with respect to the Thurston topology. This implies that there exists n 1 ∈ N such that Λ n ∈ U B(x,ε) for n ≥ n 1 and hence Λ n ∩B(x, ε) = ∅ for n ≥ n 1 . This gives a contradiction to the relation (2) which implies that Λ ′ ⊂ Λ. Now note that there is a natural identification between the subset CGL(S) and the complex of curves C(S) of S. Therefore, from Proposition 3.8, f induces an automorphism on C(S). From [4], [5] and [6] and under the hypothesis of Theorem 1.1, we obtain a homeomorphism h : S → S such that h * = f on CGL(S).
We now prove that h * = f on FGL(S). Composing f with (h * ) −1 , if necessary, it suffices to assume that f = id on CGL(S) and prove that f = id on FGL(S). To do it, we first prove two lemmas: Proof. We take a generalized pair of pants decomposition P and a maximal finite geodesic lamination Λ P such that (1) Λ ⊂ Λ P ; (2) for each component R of S − P and every pair λ 1 = λ 2 of boundary geodesics of R there is a leaf λ of Λ P contained in R which spirals in one direction about λ 1 and in the other direction about λ 2 ; (3) for every component γ of P, the leaves of Λ P spiraling about γ from different sides of γ induce opposite orientations on γ.
Such a lamination Λ P can be approximated in the Hausdorff topology and hence in the Thurston topology by a sequence C n of simple closed geodesics, where the C n are viewed as elements of CGL(S). This implies that f (Λ P ) = Λ P . Indeed, C n → Λ P with respect to the Hausdorff topology and hence with respect to the Thurston topology. Therefore f (C n ) → f (Λ P ) with respect to the Thurston topology. This implies, since f is the identity on CGL(S), that C n → f (Λ P ) with respect to the Thurston topology. From Lemma 4.1 we deduce that f (Λ P ) ⊂ Λ P . But from Lemma 3.7 the laminations Λ P and f (Λ P ) have the same length. Therefore f (Λ P ) cannot have a smaller number of leaves than Λ P . Therefore f (Λ P ) = Λ P .
We can find a surface Q ⊂ S with geodesic boundary containing the lamination Λ and which is the closure of a component of S − P such that Q is one of the following types: (i) a torus with one boundary component, with ∂Q = γ 1 or ∂Q = γ 2 ; (ii) a pair of pants with no cusps, with γ 1 ∪ γ 2 ⊂ ∂Q; (iii) a generalized pair of pants with a single cusp with ∂Q = γ 1 ∪ γ 2 .
Also, if γ is an open simple geodesic of S, we denote by γ the closure of γ in S. Obviously γ is a geodesic lamination and consists of γ and two or one additional simple closed geodesics on which γ is spiraling about. We note that we can talk about f ({γ}) and not about f ({γ}) since γ is a lamination but γ is not. }, where β 2 is the unique geodesic in Q spiraling about γ 1 in both directions and from the same side of γ 1 and such that β 2 ∩ γ = ∅. Now, we may find an open geodesic α, as well as, a closed geodesic γ ′ 1 , such that: α is spiraling about γ 1 and γ ′ 1 ; α ∩ β 1 = ∅ but α ∩ β 2 = ∅. From (ii) we have that f ({α}) = {α} and therefore we deduce that f ({β 1 }) = {β 1 } and Claim 2 is proved.
In Case (i) we distinguish the following three subcases: (i a ) γ spirals in both directions about γ 1 from different sides of γ 1 and induces opposite orientation on γ 1 from each side; (i b ) γ spirals in both directions about γ 1 from the same side of γ 1 ; (i c ) γ spirals in both directions about γ 1 from different sides of γ 1 and induces the same orientation on γ 1 from each side.
Then, as in the proof of Lemma 4.2, we prove that γ ⊂ Q. Assume that f (Λ) = Λ ′ . From Proposition 3.8 and Corollary 3.4, we deduce that Λ ′ = {γ ′ , γ 1 }, where γ ′ spirals in both directions about γ 1 . then g is either the identity element or a hyperelliptic involution. It remains to show that a hyperelliptic involution does not induce the identity on GL(S). To see this, let ι denote the hyperelliptic involution. We note that there is a pair of pants decomposition P of S that is invariant by ι. By using various ways in which an infinite geodesic in a pair of pants spirals along the boundary components, we can complete the three curves in P into a maximal geodesic lamination Λ whose image by ι is a lamination that is different from Λ. This shows that ι does not induce the identity map on GL(S). This completes the proof of Theorem 1.1.