Teichmüller Space and Fenchel-Nielsen Coordinates

Transcription

1 Teichmüller Space and Fenchel-Nielsen Coordinates Nathan Lopez November 30, 2015 Abstract Here we give an overview of Teichmüller space and its realization as a smooth manifold through Fenchel- Nielsen coordinates. Some basic notions from hyperbolic geometry are covered before defining hyperbolic structures of surfaces S of genus g 2 and the Teichmüller space T (S) of S itself. Next, Fenchel-Nielsen coordinates are constructed and used to endow T (S) with a smooth manifold structure that is unique up to diffeomorphism. The paper concludes with recent developments pertaining to these concepts. 1 Introduction The story of Teichmüller theory begins, as many mathematical stories do, with Riemann. In trying to associate a surface to multi-valued functions arising in complex analysis, Riemann formulated the idea of a Riemann surface. Further, he attempted to study the moduli space of such surfaces, and was mostly stymied by a lack of rigorous definitions. In the mid 20 th century, Oswald Teichmüller introduced the deformation space of marked Riemann surfaces, to which his name is now attached. Riemann s intuitive notions of moduli spaces were captured precisely by Teichmüller in terms of complex geometry. He defined a topology on Teichmüller space induced by the so-called Teichmüller metric, and used this to show that Teichmüller space has the structure of a complex manifold. In the following pages, we study Teichmüller space from a less analytic yet more geometric perspective. We begin by covering only the necessary ingredients from hyperbolic geometry so that we may understand the definition of a hyperbolic structure on a surface S with genus g 2. Afterwards, Teichmüller space itself is defined first as the set of all hyperbolic structures up to a certain equivalence, and second as the set of discrete, faithful representations of the surface group of S, up to conjugation. Next, the coordinates of Fenchel and Nielsen are defined and then utilized to endow the Teichmüller space with both a topological and smooth manifold structure. Finally, some recent articles covering or using these concepts are explored. 2 Preliminary Definitions We begin by considering the torus. Observe that the torus is homeomorphic to the quotient of the unit square [0, 1] [0, 1] by the group generated by the Euclidean isometries T (x, y) = (x + 1, y) and S(x, y) = (x, y + 1). 1

2 Intuitively, the maps T and S glue the two sides of the unit square. We can construct hyperbolic surfaces in an analogous way. But first, we need some definitions. 2.1 Hyperbolic Geometry Definition 2.1. The Poincaré Disc is a model of hyperbolic geometry, i.e., the set with the Riemannian metric has Gaussian Curvature 1 D = {z C z < 1} ds 2 = 4(dx2 + dy 2 (1 (x 2 + y 2 )) 2 In other words, the hyperbolic plane D is the unit circle in C with a Riemannian metric. This survey, however, is not about Riemannian manifolds, so our next definition allows us regard D simply as a metric space. Definition 2.2. Let γ : [0, 1] D be a piece-wise differentiable path. The hyperbolic length of γ is defined L(γ) = ds. Furthermore, given w, z D, the hyperbolic distance from w to z is defined d(w, z) = inf{l(γ) γ(0) = w, γ(1) = z}. Philosophically, the foundation of geometry is a set of points and a notion of lines. Now that we have our points, what do our lines look like? Definition 2.3. A geodesic is a piece-wise differentiable curve γ : [0, 1] D such that for any 0 a < b 1 the distance from γ(a) to γ(b) is realized by γ. These geodesics are the straight lines of the hyperbolic plane, in the sense that they realize the hyperbolic distance between any two points. However, in our model D, geodesics don t necessarily look straight. Indeed, there are two classes of geodesics: they are either diameters of D or semi-circles that intersect S 1 at right angles. γ 2

3 To complete the analogy of the Euclidean torus above, we need a notion of hyperbolic isometries. Definition 2.4. An injective map f : D D is an orientation-preserving isometry of D if f (as a map f : R 2 R 2 ) is differentiable For every z D det(d z f) > 0 For every w, z D d(f(w), f(z)) = d(w, z) We will usually simply say isometry instead of the full phrase orientation-preserving isometry. Since the composition of two isometries is again an isometry, the set Isom + (D) of orientation preserving isometries is a group under composition. In fact, we can identify this group with the well understood group of matrices PSL 2 (C). 2.2 Hyperbolic Structures After developing basic plane hyperbolic geometry, we are now prepared to examine surfaces endowed with hyperbolic structure. What does this mean? Definition 2.5. Let S be a topological surface. A hyperbolic structure on S is a collection {(U i, ϕ i )} i I where The collection {U i } i I is composed of open sets of S, and i I U i = S For each i I, the map ϕ i : U i D is a homeomorphism onto its image If U i U j, the transition map is an isometry of D. ϕ i ϕ 1 j : ϕ j (U i U j ) ϕ i (U i U j ) Let us now turn to our titular example of the surface of genus 2. For convenience, let s denote this surface by S. To obtain a hyperbolic structure on S, we wish to follow a parallel example to the Euclidean torus from above; that is, we need to find some polygon and a collection of isometries that glue its sides. 3

4 Consider the family of regular hyperbolic octagons with geodesic sides pictured below. As l approaches 0, the internal angle θ approaches 0 as well, since the geodesics must meet S 1 orthogonally. On the other hand, as l approaches 1, the geodesic sides become closer and closer to Euclidean straight lines. It follows that the internal angle goes to 3π/4, which is the value of the internal angle of a Euclidean regular octagon. By continuity, there must be a regular hyperbolic octagon whose internal angles are all π/4. Denote this octagon by S 8. Next, we want to glue the sides of S 8 so that the resulting quotient space is homeomorphic to S. The existence of isometries that achieve this gluing will be taken for granted, but the following diagram provides a schematic for how the gluing works and why the resulting surface is S. 4

5 Denote the group generated by the isometries pairing the sides of S 8 by Γ. It can be shown that Γ acts freely and properly discontinuously on D, a consequence of which is that the projection π : D D/Γ = S is a covering map. Thus, each point of S has an evenly covered neighborhood, and we can use these neighborhoods to define a hyperbolic structure on S. We may as well think of the points of S as being points in S 8. We consider the following cases: 1. If p is in the interior of S 8, then we can choose any neighborhood U of p that is also contained in the interior, and the chart map will be (U, Id). 2. If p is on one of the sides of S 8, but is not a vertex, then there is some isometry g Γ such that g(p) is also on a side of S 8. Choose an r > 0 sufficiently small so that the balls B r (p) and B r (g(p)) do not intersect a vertex of S 8. Then, we can produce a chart for p by setting U = (B r (p) B r (g(p))) S 8 and by defining { x if x B r (g(p)) ϕ(x) =. g(x) if x B r (p) See the figure below. 3. Finally, consider the case in which p is a vertex of S 8. Notice that under our prescribed gluing, each vertex is identified. Thus, a neighborhood U of p is the union of the wedges making up the corners of S 8. To find a map ϕ to pair with U, consider mapping each vertex to the origin by an isometry g. It is a fact that each of these isometries is conformal, so the angle of each wedge is preserved. Thus, the map ϕ can be defined as a piecewise map with 8 cases, where each wedge is mapped to the origin (see the illustration below). 5

6 2.3 Teichmüller Space Finally we can define Teichmüller space. Definition 2.6. Let S be a closed topological surface of genus g 2. The Teichmüller space of S is the quotient T (S) = {(X, f)} / where X is S equipped with a hyperbolic structure, f : S X is a homeomorphism called the marking, and (X, f) (Y, g) if and only if there is an isometry ϕ : X Y such that ϕ f is homotopic to g. Intuitively, we can think of a surface S as being naked, and T (S) as its closet. Each point (X, f) is an outfit, with X being the clothes and f the instructions for how to put the clothes on. Two outfits (X, f) and (Y, g) are considered the same if one can be obtained from the other if we can deform one into the other by twisting a collar, for instance. To continue with our previous example, we saw that the octagon S 8 gives a hyperbolic structure on S, so it defines a point in T (S). A similar construction with a decagon S 10 can be carried out to endow S with another hyperbolic structure. These two structures define distinct points in T (S), though, because S 8 and S 10 are not isometric. One can check this by considering the simple closed curve γ on S 8 depicted below. Explicitly, γ is chosen so that it intersects the midpoint of two corresponding sides of S 8 and intersects those sides at right angles. A long and involved hyperbolic trigonometry calculation shows that if there were an isometry f : S 8 S 10, then the image f(γ) could not be simple. But f is supposed to be an isometry, so this contradiction shows that S 8 and S 10 define distinct points of T (S). 3 Fenchel-Nielsen Coordinates The above definition of Teichmuüller space is quite geometric in origin. algebraically focused definition is An equivalent, yet more more Definition 3.1. Let S be a closed topological surface of genus g 2. The Teichmüller space of S is the quotient T (S) = DF(π 1 (S), P SL 2 (R)) / P SL 2 (R). That is, T (S) is the set of discrete and faithful representations of π 1 (S) into P SL 2 (R), up to conjugation. 6

7 The equivalence of these two definitions is a consequence of the Cartan-Hadamard theorem. This definition provides insight into the structure of T (S). Since DF(π 1 (S), P SL 2 (R)) Hom(π 1 (S), P SL 2 (R)) = P SL 2 (R) 2g we can see that T (S) inherits the quotient topology from P SL 2 (R) 2g. However, even more can be said about the structure of T (S). Using Fencehl-Nielsen coordinates, which we aim to define in this section, we will be able to show that T (S) carries the structure of a smooth manifold. For a surface S of genus g, we would like to define a homeomorphism F N : T (S) R 3g 3 + R 3g 3, where the first 3g 3 coordinates correspond to length parameters and the last 3g 3 coordinates are known as twist parameters. To understand this map, we introduce a few more concepts. 3.1 Pairs of Pants and Length Parameters We attempt first to decompose our surface S through the notion of a multicurve. A multicurve µ is a set of pairwise distinct and pairwise disjoint (up to homotopy) curves on S. A multicurve can contain at most 3g-3 curves, given that S has genus g, and a multicurve that attains this maximum is called a pants decomposition of S. If µ is a pants decomposition of S, then S \ µ has 2g 2 connected components and the closure of each component is homeomorphic to a sphere with three boundary components. That is, each connected component is a pair of pants. A specific pants decomposition of our surface S is depicted below. If we let P denote a single pair of pants, we can define Fenchel-Nielsen coordinates for T (P ) as a stepping stone towards general Fenchel-Nielsen coordinates. To proceed, we need a result from hyperbolic geometry concerning right angled hexagons. Lemma 3.2. Let a, b, c R +. Then there exists a unique, up to isometry, right angled hyperbolic hexagon with alternating edge lengths given by a, b, and c (see the figure below). Proposition 3.3. Let γ 1, γ 2, and γ 3 denote the boundary curves of the pair of pants P. The map F N : T (P ) R 3 + given by F N(X) = (L X (γ 1 ), L X (γ 2 ), L X (γ 3 )), where L X is the hyperbolic length with respect to the structure X, is a homeomorphism. 7

8 Proof. We illustrate a sketch of the proof. First, if (a, b, c) R 3 +, then by the lemma there exists a unique right angled hyperbolic hexagon H with alternating side lengths a/2, b/2, and c/2. Gluing two copies of H along the other sides produces a hyperbolic structure on P. Therefore, F N is onto. Let X be a structure on P such that F (X) = (a, b, c). For each pair of distinct curves γ i and γ j, there is a unique geodesic arc A ij connecting γ i to γ j that intersects each curve orthogonally. Then, X\(A 12 A 13 A 23 ) has two connected components H 1 and H 2. Notice that the closure of each H i is homeomorphic to a hyperbolic hexagon, so by the lemma each is unique up to isometry. It follows that X is unique as well, so that F N is injective. Finally, we need to show continuity. If we use our second definition of T (P ), then each point is given by a representation ρ : π 1 (P ) P SL 2 (R). Moreover, each simple closed curve γ is a word in the generators of π 1 (P ). Thus, we can define the length function evaluated on γ as the translation length of the corresponding element of P SL 2 (R). That is, ( ) Tr(ρ(γi )) L X (γ i ) = 2arccosh 2 and this is continuous with continuous inverse. 3.2 Twist Parameters and General Fenchel-Nielsen Coordinates Next, we tackle the general case. Let S, as always, be a surface of genus g 2 and fix a pants decomposition µ = {γ 1,..., γ 3g 3 } of S. Define the Fenchel-Nielsen coordinates of T (S) as a map F N : T (S) R 3g 3 + R 3g 3 by requiring that F N(X) = (L X (γ 1 ),..., L X (γ 3g 3 ), τ 1 (X),..., τ 3g 3 (X)). Having defined the first 3g 3 length parameters, we turn to defining the twist parameters τ i (X). Fix a boundary curve γ i as well as an orientation for γ i, and note that it is contained in a unique component C i of X \ j i γ j. The closure of C i is either (1) a torus with one boundary component or (2) a sphere with four boundary components. In either case, let α i be a curve completely contained in C i that intersects γ i minimally. Additionally, in each pair of pants S i \ γ i, let {β i,j } j be the collection of geodesic (with respect to the structure X) curves that are entirely contained in each pair of pants, have endpoints of γ i, and intersect γ i orthogonally. Note that in the first case above there is only one curve β i,1, and in the second there are two curves {β i,1, β i,2 }. For simplicity, we consider the first case in which S i is a torus with one boundary component. 8

9 In this case, we can homotope α i onto a curve α i completely contained in γ i β i,1. After choosing one of the two basepoints p γ i β i,1, define τ i (X) as the signed length of the segment of α i that runs along γ i and contains p. The sign of τ i (X) is positive if α i runs along the orientation of γ i, and negative otherwise. We can state our main theorem. Theorem 3.4. The map F N : T (S) R 3g 3 + R 3g 3 defined at the beginning of this subsection is a homeomorphism. We already saw that the length parameters are continuous in Proposition 3.3. Moreover, the construction of the twist parameters shows that each τ i (X) is defined in terms of length functions, so they are also continuous. Finally, F N is bijective because given a tuple (L 1,..., L 3g 3, τ 1,..., τ 3g 3 ) R 3g 3 + R 3g 3, one can construct 2g 2 pairs of pants, with length components determined by the L i, and then glue them all together according to the twist parameters τ i. 3.3 Smooth Structure of T (S) Choose a pants decomposition µ of S. The Fenchel-Nielsen coordinates allow us to define a smooth manifold structure on T (S) by pulling back the natural structure on R 6g 6. Explicitly, since (R 6g 6, Id) defines an atlas on R 6g 6, it follows that (F N 1 (R 6g 6 ), Id F N) = (T (S), F N) defines a 6g 6-dimensional atlas on T (S). Note however, that this smooth structure depends a priori on the pants decomposition µ. In fact, two distinct pants decompositions yield the same smooth structure, up to diffeomorphism. We proceed to explain why this is true, leaving most statements unproven. Let µ = {γ 1,..., γ 3g 3 } be a pants decomposition of S, and for a fixed γ i consider the two cases for the closure of the component C i discussed above. Recall that α i represented a minimally intersecting curve of γ i. If we exchange the roles of γ i and α i in C i, we obtain a new pants decomposition of S. Call these moves an A-move and an S-move, respectively. It can be shown that if the pants decomposition µ can be obtained from µ by either an A-move or an S-move, then the change of coordinates between the Fenchel-Nielsen coordinates induced by µ and µ is a diffeomorphism. Denote by P(S) the graph whose vertex set is the set of all pants decompositions of S, and place an edge between the pants decompositions P and P if one can be obtained from the other by either an A-move or an S-move. It is a non-trivial fact that this graph is connected. This, combined with the fact that the composition of two diffeomorphisms is again a diffeomorphism, implies that any two pants decompositions µ and µ produce the same smooth structure for T (S), up to diffeomorphism. 9

10 4 Other Deformation Spaces Recall Definition 3.1 of Teichmüller space: T (S) = DF(π 1 (S), P SL 2 (R)) / P SL 2 (R). Such a quotient of representations is called a deformation space. However, there is nothing restricting us to studying representations into P SL 2 (R). In fact, in a recent paper by Gongopadhyay and Parsad, the authors study the space T (S) = DF(π 1 (S), SU(3, 1)) / SU(3, 1). By decomposing their surface into a pairs of pants, Gongopadhyay and Parsad are able to come up with coordinates analogous to traditional Fenchel-Nielsen coordinates as described above. Theorem 4.1. (Gongopadhyay and Parsad): Let S be a closed surface of genus g with a pair of pants decomposition C = {γ j } 3g 3 j=1. Let ρ : S SU(3, 1) be a tame representation. Then 30g 30 real numbers are needed to specify ρ in the deformation space DF(π 1 (S), SU(3, 1))/SU(3, 1) In the theorem, a representation ρ is called tame if the resulting (0, 3) groups from the given pants decomposition are all non-singular in SU(3, 1). References [1] J. Aramayona, Hyperbolic Structures on Surfaces. Geometry, Topology and Dynamics of Character Varieties, World Scientific 23 (2012) [2] K. Gongopadhyay and S. Parsad, On Fenchel-Nielsen Coordinates of Surface Group Representations into SU(3, 1). (2014) [3] A. Hatcher, Pants Decompositions of Surfaces. hatcher/papers/pantsdecomp.pdf, (2000) [4] L. Ji and A. Papadopoulos, Historical Development of Teichmüller Theory. Archive for History of Exact Sciences, Volume 67, Issue 2, pp , (2013) [5] C. Series, Hyperbolic Geometry, MA masbb/papers/ma448.pdf, (2008) 10