# Dual Teichmüller spaces.

###### Abstract

We describe two spaces related to Riemann surfaces — the Teichmüller space of decorated surfaces and the Teichmüller space of surfaces with holes. We introduce simple explicit coordinates on them. Using these coordinates we demonstrate the relation of these spaces to the spaces of measured laminations, compute Weil-Petersson forms, mapping class group action and study properties of lamination length function. Finally we use the developed technique to construct a noncommutative deformation of the space of functions on the Teichmüller spaces and define a class of unitary projective mapping class group representations (conjecturally a modular functor). One can interpret the latter construction as quantisation of 3D or 2D Liouville gravity. Some theorems concerning Markov numbers as well as Virasoro orbits are given as a by-product.

## 1 Introduction.

The main philosophical aim of the paper is to formulate two problems concerning Teichmüller spaces of Riemann surfaces with holes.

The first problem is to describe explicitly a kind of Fourier transform between the spaces of functions on two slightly different versions of Teichmüller spaces.

The second problem is to deform (quantise) the algebra of functions on Teichmüller space in a direction prescribed by the Weil-Petersson Poisson bracket and compatible w.r.t. homotopy classes of mappings between surfaces. In particular w.r.t. the mapping class group action.

The solution for the second problem is given in this version of the text, however we do not give here detailed proofs and examples which is postponed to a subsequent paper. Concerning the first problem, we just try to give arguments in favor of existence of a solution and emphasise its importance.

The technique used in the article unifies the Thurston approach to Riemann surfaces, such as measured laminations on the one hand, and mathematical physics such as modular functors of conformal field theory on the other. One metaresult important for us was the construction of a bridge between these domains.

However the article does not contain philosophical discussions except for a few remarks. In the main part of the text we give definitions and prove theorems (which can be considered as preparatory in the spirit of the problems described above, but we hope that they have some independent interest as well). The main ones of them are:

We define explicitly simple global coordinates on the Teichmüller spaces of Riemann surfaces with holes. We describe Penner’s coordinates on the decorated Teichmüller spaces. We give explicit formulae for the action of the mapping class group as well as for the Weil-Petersson Poisson bracket on the former space and for the Weil-Petersson degenerate symplectic structure for the latter one. We show using the coordinates that these spaces have natural ”scaling” limits to the space of measured laminations with closed (resp. compact) support. We give an elementary proof of continuity of the lamination length function and the lamination intersection number. As a by-product of the latter statement we prove some continuity theorem concerning Markov numbers. We show also explicitly compatibility of the length and the intersection index functions at the limit when Teichmüller spaces go to the respective laminations. We describe Bers’s coordinates on the simplest Virasoro orbit and as another by-product we compute the Kirillov–Kostant Poisson bracket in terms of these coordinates. Finally we give an explicit construction of noncommutative deformation of the space of functions on the Teichmúller spaces of Riemann surfaces with holes depending on a quantisation parameter and show an amazing symmetry between deformations corresponding to the parameters and .

We tried to make the paper to be self-contained and available for a wide class of readers. Therefore we have included many known results. Some of them are provided with a few line proofs in the spirit of the paper. The other proofs are left for the interested reader as easy exercises. Some slightly more complicated results in the two final sections are provided with references to a proof.

For a nonrigorously minded reader we remark that the quantisation procedure gives Hilbert spaces which can be interpreted either as the space of conformal blocks of Liouville gravity theory in two Euclidean dimensions or as the space of states in 3D quantum gravity, as it follows form the ideas of [14], [13]. The symmetry is rather similar to the one observed in [15]. However we omit (except for a few sentences at the end of some sections) the discussion of this point of view here since we can hardly imagine arguments making something more out of this statement than just a definition.

## 2 Graphs and surfaces.

In this section we shall give a brief description of relations between surfaces and fat graphs. These relations exist only for surfaces with the number of holes , genus or with at least 3 holes and genus . Such surfaces will be called hyperbolic. All graphs considered in thesequel are supposed to be finite.

Recall that a fat graph is an unoriented graph s.t. for each vertex the cyclic order of ends of edges incident to the vertex is given.

One can imagine a fat graph as a graph with edges being narrow bands. (It is where the attribute fat comes from.) A graph drawn on an oriented surface acquires a fat graph structure given by, say, a counterclockwise ordering of the ends of edges at each vertex.

Let us say that an oriented path on a fat graph turns left at a vertex if we come to the vertex along the end of an edge which is precedent to the one we come out w.r.t. the cyclic order.

On a fat graph one can well define a distinguished set of closed paths called faces. A face is a path s.t. being oriented it turns always left at each verex or always right.

Denote by and the sets of vertices, edges and faces of , respectively.

We can obtain a smoothable surface from a fat graph by taking a disk for each face and gluing its boundary to the graph along the face. If we have taken the set of edges and vertices of a polyhedron in as a graph with a natural fat structure, we recover by this procedure the original polyhedron, the faces of the graph being correspondent to the faces of the polyhedron. And this is the reason to use the term face in this context. We can use annuli instead of disks and get a surface with boundary. The surface can be obviously retracted onto .

One can give a purely combinatorial description of a fat graph. Let be the set of ends of edges of a fat graph (or what is the same, the set of oriented edges). Define an involution acting on which maps an end of an edge to the opposite end of the same edge (resp. reverse the orientation of the edge). The fat structure induces a permutation of the same set which maps an end of an edge to the next end of an edge w.r.t. the cyclic order at the corresponding vertex. It is obvious that this gives a one-to-one correspondence between fat graphs and pairs of permutations on finite sets, s.t. is an involution without fixed points. In these terms faces correspond to orbits of in .

Introduce some notation useful for the sequel. Let be an end of an edge. Then let

We say that , where and if the orientation of agrees with the counterclockwise orientation of . Denote by the edge from corresponding to .

Note that for a fat graph one can define the dual graph with vertices, edges and faces replaced by faces, edges and vertices of , respectively. In terms of permutations the dual graph corresponds to the pair , where acting on the same set.

Now show that any hyperbolic surface can be obtained as for some fat graph . The construction becomes more transparent if we imagine the holes having zero size, i.e., just as punctures. Any surface can be cut into topologically trivial pieces by a number of curves going from puncture to puncture which are self- and mutually nonintersecting, mutually homotopically nonequivalent and nonshrinkable to punctures. We can always take enough curves to make the pieces simply connected. The resulting set of curves is a graph with vertices at the punctures and a natural fat structure given by the orientation of the surface. The desired graph is dual to this one. One can easily check by drawing pictures that the surface is isomorphic to the surface we have started with. If we now take one point inside each piece and for each cut draw a segment with ends in the chosen points intersecting only this cut we obtain the graph together with the homotopy class of its embedding into .

A maximal system of such curves (it always exists) cuts the surface into triangles and the corresponding graphs turn out to have three ends of edges incident to each vertex (3-valent graphs). In the sequel we mostly consider such kind of graphs. One can easily check by computing the Euler characteristics that a 3-valent graph has edges vertices and faces, where and are the genus and the number of holes of the surface , respectively.

Denote by the set of graphs, corresponding to a given surface and by the set of graphs together with their embeddings into considered up to homeomorphisms of homotopy equivalent to the identity. The former set is finite and the latter is obviously infinite. The mapping class group acts naturally on with as a quotient. The subset of (resp. consisting of three-valent graphs is denoted by (resp. It is obviously stable w.r.t. .

If a graph has a symmetry, it acts obviously on the corresponding elements of . One can show that any two elements of are connected by a sequence of flips and graph symmetries. In particular, the action of any element of the mapping class group on can be represented by a sequence of flips and symmetries. In more scientific words and sequences of flips and symmetries constitute a groupoid containing the mapping class group as the greatest subgroup.

Note that if is an unramified -fold covering then one can obviously construct a graph corresponding to starting from a graph corresponding to . (This graph is just the full inverse image of in . has edges, vertices and faces. Here is the number of orbits of the covering monodromy around the face . There is a natural mapping from the edges, vertices and faces of to the edges, vertices and faces of , respectively, which we shall denote by the same letter .

The mapping class group obviously acts on the set of unramified -fold coverings of . A stabiliser of a covering in we call a congruence subgroup w.r.t. and denote by . is obviously a finite index subgroup in

Call a three-valent fat graph regular if it has no edges with coinciding ends, any two edges have no more than one common vertex and any edge separates two different faces. Not all surfaces can be represented by regular graphs. The only reason to introduce this class of graphs is because usually all constructions and formulae are more simple for them. However any nonregular graph can be covered by a regular one, and usually one can easily derive formulae for nonregular graphs starting from those for regular graphs by passing to such a covering.

## 3 Laminations.

Taking into account that the reader may not be familiar with the Thurston’s notion of a measured lamination [11], we are going to give all definitions here in the form, which is almost equivalent to the original one (the only difference is in the treatment of the holes and punctures), but more convenient for us. The construction of coordinates on the space of laminations we are going to describe is a slight modification of Thurston’s ”train tracks” ([11], section 9 and [8]).

It seems worth mentioning here, that the definitions of measured laminations are very similar to the definitions of the singular homology groups, and is in a sense an unoriented version of the latter ones.

There are two different ways to define the notion of measured laminations for surfaces with boundary, which are analogous to the definition of homology group with compact and closed support, respectively.

### 3.1 Bounded measured laminations.

#### Definition.

Rational bounded measured lamination on a 2-dimensional surface is a homotopy class of a collection of finite number of self- and mutually nonintersecting unoriented closed curves with rational weights and subject to the following conditions and equivalence relation.

1. Weights of all curves are positive, unless a curve surrounds a hole.

2. A lamination containing a curve of weight zero is considered to be equivalent to the lamination with this curve removed.

3. A lamination containing two homotopy equivalent curves with weights and is considered as equivalent to the lamination with one of these curves removed and with the weight on the other.

The set of all rational bounded laminations on a given surface is denoted by This space has a natural subset, consisting of laminations with integral weights. The set of such laminations is denoted by . Denote by and the subspaces, consisting of laminations without curves surrounding holes.

Remark. Any rational bounded measured lamination can be represented by a collection of curves. Any integral lamination can be represented by a finite collection of curves with weights or on some curves surrounding holes.

#### Construction of coordinates.

Suppose we are given a three-valent fat graph . We are going to assign, for a given lamination, rational numbers on edges and show, that these numbers are good coordinates on the space of laminations.

#### Reconstruction.

Now we need to prove that these numbers are coordinates indeed. For this purpose we just describe an inverse construction which gives a lamination, starting from numbers on edges.

First of all note, that if we are able to reconstruct a lamination, corresponding to a set of numbers , we can do it as well for the set and for any rational and . Indeed, multiplication of all numbers by can be achieved by multiplication of all weights by and adding is obtained by adding loops with weight around each hole. Therefore, we can use these possibilities to reduce our problem to the case when are positive integers and any three numbers on edges incident to each vertex satisfy triangle and parity conditions

(3) | |||

(4) |

Now the reconstruction of the lamination is almost obvious. Draw lines on the th edge and connect these lines at vertices in a nonintersecting way (fig.3.1), what can be done unambiguously.

Fig. 5

#### Graph change.

The constructed coordinates on the space of laminations is related to a particular choice of the three-valent graph. The following formulae describe the change of coordinates under a flip of an edge of the graph.

Fig. 6

(Only the changing part of the graph is shown here, the numbers on the other edges remain unchanged.)

### 3.2 Unbounded measured laminations

#### Definition

Rational unbounded measured lamination on a 2-dimensional surface with boundary is a homotopy class of a collection of finite number of nonselfintersecting and pairwise nonintersecting curves either closed or connecting two boundary components (possibly coinciding) with positive rational weights assigned to each curve and subject to the following equivalence relations:

1. A lamination, containing a curve retractable to a boundary component is equivalent to the lamination with this curve removed.

2. A lamination containing a curve of zero weight is considered to be equivalent to the lamination with this curve removed.

3. A lamination containing two homotopy equivalent curves of weights and , respectively, is equivalent to the lamination with one of these curves removed and with the weight on the other.

The set of all rational unbounded laminations on a given surface is denoted by . This space has a natural subset, representable by collections of curves with integral weights. This space is denoted by .

Remark. Any rational unbounded measured lamination can be represented by a collection of no more than curves (for Euler characteristics reasons). Any integral lamination can be represented by a finite collection of curves with unit weights.

For any given lamination, fix orientations of all boundary components but those nonintersecting with curves of the lamination. Denote the space of rational (resp. integral) laminations equipped with this additional structure by (resp. ).

#### Construction of coordinates

Suppose we are given a three-valent fat graph . We are going to assign for a given element of the space a set of rational numbers on edges, and show that these numbers are good global coordinates on this space.

Straightforward retraction of an unbounded lamination onto is not good because some curves may shrink to points or finite segments. To avoid this problem, let us first rotate each oriented boundary component infinitely many times in the direction prescribed by the orientation as shown on fig. 3.2.

Fig. 7

We can mark a finite segment of each nonclosed curve in such a way that each of two unmarked semiinfinite rays goes only around a single face and therefore are never right or left handed. Therefore only the finite marked parts of curves contribute to the numbers on the edges.

#### Reconstruction

Now we need, as in the bounded case, to prove that these numbers are indeed coordinates, what we shall do as well by describing an inverse construction. Note that if we are able to construct a lamination corresponding to the set of numbers , we can equally do it for the set for any rational . Therefore we can reduce our task to the case when all numbers on edges are integral. Now draw -infinitely many lines along each edge. In order to connect these lines at vertices we need to split them at each of the two ends into two -infinite bunches to connect them with the corresponding bunches of the other edges. Let us make it at the -th edge, such that (resp. ), in such a way, that the intersection of the right (resp. left) bunches at both ends of the edge consist of lines (resp. lines). Here the left and the right side are considered from the centre of the edge toward the corresponding end. The whole procedure is illustrated on fig. 3.2. The resulting collection of curves may contain infinite number of curves surrounding holes, which should be removed in accordance with the definition of an unbounded lamination.

Fig. 9

Note that although we have started with infinite bunches of curves the resulting lamination is finite. All these curves glue together into a finite number of connected components and possibly infinite number of closed curves surrounding punctures. Indeed, any curve of the lamination is either closed or goes diagonally along at least one edge. Since the total number of pieces of right or left handed curves is finite the resulting lamination contain no more than this number of connected components. (In fact the number of connected components equals provided all numbers are all nonpositive or all nonnegative.)

#### Graph and orientation changes.

Here is the transformation law for the constructed coordinates for a flip of an edge for a simple graph.

Fig. 10

(Only changing part of the graph is shown here, the numbers on the other edges remain unchanged.)

One can write down explicitly what happens to the coordinates when one changes the orientation of a hole. Since the formulae are relatively complicated we postpone them to the sixth section.

#### Relations and common properties of and .

1. Since the transformation rules for coordinates (3.1) and (3.2) are continuous w.r.t. the standard topology of the coordinates define a natural topology on the lamination spaces. One now can define the spaces of real measured laminations (resp. bounded and unbounded) as a completion of the corresponding spaces of rational laminations. These spaces are denoted as , and , respectively. Of course we have the coordinate systems on these spaces automatically.

Note that to define real measured laminations it is not enough just to replace rational numbers by real numbers in the definition of the space of laminations. Such definition would not be equivalent to the one given above since a sequence of more and more complicated curves with smaller and smaller weights may converge to a real measured lamination. Thurston in [11] defined real measured laminations directly as transversely measured foliation of closed submanifolds. It seems to us that our definition is more convenient for practical computations although it does not work well for surfaces without boundary.

2. An unbounded lamination is integral if and only if it has integral coordinates. A bounded lamination is integral if and only if it has integral coordinates and the sum of three numbers on edges incident to each vertex is even.

3. If is an unramified covering then we can define an inverse image of a lamination . For a rational the curves of are just full inverse images of the curves of with the same weights as on the respective curves of . This mapping can be obviously extended to all laminations. The analogous mapping can be analogously defined for the spaces of unbounded laminations.

Note that the graph coordinates of a lamination w.r.t. to the graph are just pullbacks of the graph coordinates of the lamination w.r.t. , i.e.,

The constructed mappings and are embedings.

4. Denote the closure of in by . We have the following commuting diagram of natural mappings commuting with the action of the mapping class group:

(11) |

The projection forgets the curves surrounding holes; the projection (resp ) is given by the total weights of ends of curves entering the hole (resp. taken with minus sign for the case of if the orientation of the hole is opposite to the one induced by the orientation of the surface); and are the canonical projections on the quotient by the group acting by changing orientation of the holes on and by changing sign of the standard coordinates on , respectively; is given by the weights of the curves surrounding holes; is a family of embeddings characterised by the condition that for any . The image of coincides with the kernel of and with the stable points of the action.

In coordinates the mapping is given by

(12) |

where and are the coordinates on and , respectively, w.r.t. the same graph . By and we mean the numbers assigned to the corresponding unoriented edges.

The mapping is given by

(13) |

The mapping (resp. ) is given by

(14) |

## 4 Teichmüller spaces.

The Teichmüller space (resp. Moduli space ) of a closed surface is the space of complex structures on modulo diffeomorphisms homotopy equivalent to the identity (resp. modulo all diffeomorphisms). The extension of these notions to surfaces with boundary depends on the condition that one imposes on the behaviour of the complex structure at the boundary of the surface. The most traditional definition considers only complex structures degenerating at the boundary and such that a tubular neighbourhood of each boundary component is isomorphic as a complex manifold to a punctured disc. (Such kind of singularity is called puncture.) We denote the corresponding Teichmüller and moduli spaces by the same letters and , respectively. We describe two other modifications of Teichmüller spaces and give explicit parameterisations of them. But before we just recall some basic facts about relations between complex structures, constant negative curvature metrics and discrete subgroups of the group . For more details we recommend the reviews [9].

According to the Poincaré uniformisation theorem any complex surface can be represented as a quotient of the upper half plane by a discrete subgroup of its automorphism group (sometimes called the Möbius group) of real matrices with unit determinant considered up to the factor . The group is canonically isomorphic to the fundamental group of the surface . is defined by the complex structure of the surface up to conjugation by an element of . Therefore we get an embedding . This embedding has the following properties:

1. The image of any loop is a matrix with one or two real eigenvectors. (Such elements of are called parabolic and hyperbolic, respectively.)

2. Parabolic elements correspond to loops surrounding punctures only.

The proof of these well known properties will in particular follow from the construction of the parameterisations.

On there exists a unique -invariant Riemann curvature metric. It induces a metric on . Since this metric is of negative curvature, any homotopy class of closed curves contains a unique geodesics. Homotopy classes of closed curves on a surface are in one-to-one correspondence with the conjugacy classes of its fundamental group . Denote by an element of and by the length of the corresponding geodesics. Then a simple computation shows, that

(15) |

where and are the eigenvalues of the element of corresponding to . This number is obviously correctly defined, i.e., it does not depend on the choices of particular representation of , of a particular element of , representing a given loop and of a particular matrix representing an element of . This formula implies that the length of a geodesics surrounding a puncture is zero. Note that taking curvature to be is equivalent to the demand that the curvature is negative and constant and areas of ideal triangles are equal to , which normalisation condition is more convenient practically.

### 4.1 Teichmüller space of surfaces with holes .

#### Definition.

There is another condition one can impose on the behaviour of complex structure in a vicinity of the boundary and still get a finite dimensional moduli space. Demand that a boundary component be either a puncture or the complex structure is nondegenerate at the boundary. A boundary of the latter type is called a hole. A neighbourhood of a hole is isomorphic as a complex manifold to an annulus. The corresponding moduli space is denoted by .

For our purposes it is more convenient to introduce another space. is the space of complex structures on together with orientations of all holes. (By orientation of a hole we mean the orientation of the corresponding boundary component.) Note that, although it is not a priori obvious, this space possesses a natural topology in which it is connected.

#### Construction of coordinates.

Let be a three-valent graph, corresponding to a surface . For any point of we are going to describe a rule for assigning a real number to each edge of . The collection of these numbers will give us a global parameterisation of .

For simplicity consider first the case, when all boundary components are holes. Draw a closed geodesics around each hole and cut out cylinders by them. We thus get a surface with geodesic boundary. Then cut the surface by the edges of the dual graph into hexagons. (These edges are not necessarily geodesic though one can suppose them to be.) Take an edge and two hexagons incident to it and lift the resulting octagon to the upper half plane . The octagon has four geodesic sides facing holes. Continue these geodesics up to the real axis. Now, the orientations of the holes induce the orientations of the geodesics. Using these orientations choose one of the two infinities of each geodesics, say, the end. We obtain therefore four points on the real axis, or to be more precise, on . Note, that the four geodesics do not intersect on , and therefore the cyclic order of the constructed points on does not depend on the point of we have started with. Among the constructed four points there are two distinguished ones which originate from the geodesics connected by the edge we have started with. Using the action of the Möbius group on the upper half plane, we can shift these two points to zero and infinity, respectively, and one of the remaining points to . And now finally assign to the edge the logarithm of the coordinate of the fourth point. (Of course this forth coordinate is nothing but a suitable cross-ratio of those four points.)

Fig. 16

Note that if we have punctures instead of some holes it does not spoil the construction. In this case some edges of the considered hexagons shrink to points, the corresponding geodesics on the upper half plane shrink to points on the real axis and no orientation is necessary to choose between their ends.

#### Reconstruction.

Our goal now is to construct a surface starting from a three-valent fat graph with real numbers on edges. First of all give a simple receipt how to restore orientations of the boundary components from these data: The orientation of a boundary component corresponding to a face is just induced from the orientation of the surface (resp. opposite to the induced one) if the sum is positive (resp. negative). If the sum is zero, it means, that it is not a boundary, but a puncture.

Construction of the surface itself can be achieved in two equivalent ways. We shall describe both since one is more transparent from the geometric point of view and the other is useful for practical computations.

Construction by gluing. We are going to glue our surface out of ideal hyperbolic triangles. The lengths of the sides of ideal triangles are infinite and therefore we can glue two triangles in many ways which differ by shifting one triangle w.r.t. another along the side. The ways of gluing triangles can be parameterised by the cross-ratios of four vertices of the obtained quadrilateral (considered as points of ). For our purpose it is convenient to take as a parameter the logarithm of the cross-ratio

(17) |

where , and are coordinates of vertices of the quadrilateral, and being coordinates of the ends of the side we are gluing triangles along.

Now consider the dual graph . Its faces are triangles. Take one ideal hyperbolic triangle for each face of this graph and glue them together along the edges just as they are glued in using numbers assigned to the edges as gluing parameters.

Note that although this is not quite obvious the resulting surface is not necessarily complete. In fact it is not the original surface with the absolute boundary but only what we get out of it by cutting off annuli around holes by closed geodesics.

Construction of the Fuchsian group. We are now going to construct a discrete subgroup of starting from a graph with numbers on edges. Modify first the original graph at each vertex in the following way. Disconnect the edges at the vertex and then connect them by three more edges forming a triangle. Orient the edges of the triangle in the counterclockwise direction. Now assign to each of these edges the matrix the matrix instead of each time we go along a new edge in inverse direction w.r.t. the orientation. (The orientation of the old edges is not to be taken into account, since and therefore coincides with its inverse in the group .) In particular, if we take closed paths starting form a fixed vertex of the graph, we get a homomorphism of the fundamental group of to the group . The image of this homomorphism is just the desired group . . Now for any oriented path on this graph we can associate a matrix by multiplying consecutively all matrices we meet along it, taking . Assign to each old edge

In principle we need to prove that these two constructions are inverse to the above construction of coordinates indeed, what is almost obvious, especially for the first one. The only note we would like to make here is to show where the matrices and came from. Consider two ideal triangles on the upper half plane with vertices at the points and , respectively. Then the Möbius transform which permutes the vertices of the first triangle is given by the matrix , and the one which maps one triangle to another (respecting the order of vertices given two lines above) is given by .

#### Graph and orientation change.

Here is the transformation law for the constructed coordinates for a flip of an edge.