By Lemma 12.3, the group (Z/2Z)n acts by positive birational transformations on the moduli space XPGL2,S. We extend the positive atlas on XPGL2,S by adding co-ordinate systems obtained from the standard ones by the action of (Z/2Z)n.
1. Positive coordinate systems: definitions and the main result. — Let CS be the set of pairs (an ideal triangulation ofSconsidered modulo isotopy, a choice of orientations of boundaries of all holes of S). The direct product of the mapping class group of S and the group (Z/2Z)n acts naturally on the set CS. The coordinate systems on the space of X-laminations on S are parametrized by the set CS. So there are 2n coordinate systems corresponding to an ideal triangulation ofS.
14.1. Definition. — Let l be an X-lamination on S. A coordinate system corresponding to an ideal triangulation of Sis positive (non-negative) for the lamination l, if all coordinates of l in this coordinate system are > 0 (respectively ≥0).
14.1. Theorem.
a) There exists a dense (resp. dense open) subset in the set of all real X-laminations on S such that every lamination from it admits a non-negative (resp. positive) coordinate system.
b) Let T be an ideal triangulation of S corresponding to a non-negative coordinate system for a finite real X-lamination l. Then the isotopy class of the collection of edges of T with
> 0 coordinates is isotopic to l, and hence uniquely determined by l.
c) A positive coordinate system for an X-lamination is unique if exists.
To get the part a) of Theorem 14.1 we need the following result: for a generic real X-lamination all curves of the lamination go between the punctures. For this we use some results from the theory of Morse–Smale foliations.
2. Morse–Smale foliations and real laminations. — Let us recall the notion of a Morse–Smale foliation on S and some results about them.
Given a foliation on a punctured disc, the winding number of the foliation at the puncture is an integer given by the rotation number of the fibers of the foliation re-stricted to a small circle near the puncture. For example the radial foliation has the winding number +1, and the foliation tangent to the family of concentric circles also has the winding number +1.
Let S be a surface with holes. A Morse–Smale foliation on S is a foliation F on Sminus a finite number of points, called singular points of the foliation, with the fol-lowing structures/properties:
i) F is equipped with a real transversal measure.
ii) The fibers of F are transversal to the boundary of S.
iii) The winding number of the foliation at each singular point is < 0.
A fiber of a Morse–Smale foliation on Sis singular if it ends at a singular point of the foliation. The condition iii) implies that the number of singular fibers of a Morse–
Smale foliation ending in a singular point is finite. The condition ii) easily implies that a Morse–Smale foliation does not have fibers contractible onto the boundary.
14.1. Lemma. — The number of homotopy types of the leaves of a Morse–Smale foliation going between the boundary components is finite.
Proof. — The above remarks show that we may assume that the fibers can not be contracted to the boundary. Cutting the surface along a non-singular leaf going between the boundary components we decrease either the number of holes, or the genus. The lemma follows.
We need the following result.
14.2. Theorem. — There exists an open dense domain UMS(S)in the space of all Morse–
Smale foliations on S such that every non-singular curve of a foliation from this domain joins two boundary components.
Now let us replace the homotopic non-singular fibers of a Morse–Smale folia-tion F by a single curve with the weight provided by the transversal measure. Ac-cording to a well known theorem of Thurston, this way we get a bijection between the Morse–Smale foliations onS modulo isotopy and Thurston’s R-laminations: every R-lamination is representable as a Morse–Smale foliation on S, and this representa-tion is unique up to an isotopy. It follows from Lemma 14.1 that if F was from the setUMS(S), we will get a Thurston R-lamination lF with a finite number of curves.
3. Proof of Theorem 14.1. — Obviously c) is a particular case of b). Moreover, clearly in b) not only the set of the edges of T carrying positive coordinates, but also the set of the orientations of the holes serving as the endpoints of these edges is de-termined by l.
a) We will assume without loss of generality that an X-lamination l is defined using the orientations of the holes induced by the orientation of the surface S. The general case is reduced to this using the action of the group (Z/2Z)n changing orien-tations of the holes.
A real X-lamination is called finite if it can be presented by a finite collection l of mutually and self non-intersecting curves with real weights on them. Recall that to get an X-lamination out of such a collection of curves with real weightsl we have to
choose in addition orientations of all holes. As we already said, we do it by taking the orientations induced by the orientation of the surface S, getting an X-lamination l+. To define the X-coordinates of the finite X-lamination l+ we have to perform the following procedure at every hole of S: we replace a segment of the lamination going to the hole by a ray winding infinitely many times around the puncture in the di-rection prescribed by the orientation of S, and then retract the winded lamination to a trivalent graph on S.
Let us call finite real X-laminations such that every leaf of the lamination joins two boundary component special real X-laminations. Theorem 14.2 and Lemma 14.1 imply that the space of special finite real X-laminations is an open dense subset in the space of real X-laminations on S.
Now, given a special X-lamination l+, let us exhibit a non-negative triangula-tion ofSfor it. The curves of the finite lamination l provide a decomposition Dl ofS.
Adding a finite number of curves to this decomposition we obtain a triangulationT of Scontaining the decomposition Dl. Let Γbe the dual graph for this triangulation. We claim that it provides a non-negative coordinate system for the X-lamination l+: all coordinates assigned to the edges of Dl are > 0, while the rest of the coordinates are equal to zero. Indeed, a curve of the lamination, being retracted onto the graph Γ, will cross only one edge of Γ, the one dual to the edge of the triangulation T given by this curve. Figure 14.1 illustrates this claim, by showing the winding procedure for a curve of the lamination onStraveling between the holes. The part a) of the theorem is proved.
. .
FIG.14.1. — Calculating a coordinate of anX-lamination assigned to an edge of the dual graph.
b) Given a special real X-lamination l, there is the following correspondence Il ⊂ {edges of Γ} × {connected components of the lamination l}.
(14.1)
Namely, a pair (an edge e of Γ, a connected component α of l) belongs to Il if and only if there exists a part of α which goes diagonally along the edge e. Let us denote byπ2 the natural projection of the correspondence Il to the second factor in (14.1):
π2 :Il −→ {connected components of the lamination l}.
14.2. Lemma. — Let l be a special real X-lamination on S. Then:
a) The map π2 is surjective.
b) If π2−1(α) is a single edge e, then this edge is isotopic to an edge of the dual triangula-tion T.
c) If all coordinates of l are ≥0, then the map π2 is injective.
Proof. — a) Take a connected component β of l which is not in the image of π2. This means that there is no edge of Γ such that β goes diagonally along this edge.
Thus β can be shrunken into the boundary of a hole. Indeed, β either turns to the right at every edge ofΓ, or it turns to the left at every edge ofΓ, and thus it is isotopic to a curve which does not intersect Γ.
b) If π2−1(α) = e, then α is homotopic to a curve which intersects Γ just once.
Indeed, it either goes to the left before e and to the right aftere, or to the right before and to the left aftere. The claim follows from this.
c) Assume the opposite. Then there is a connected component α of l such that
|π2−1(α)|> 1. But this is impossible: after the curve α crossed diagonally the first edge on its way, it can not cross the other edge since all its coordinates are≥0. The lemma is proved.
14.1. Corollary. — Let l be a special real X-lamination onS. Suppose that all coordinates ofl in the coordinate system corresponding to an ideal triangulation T are ≥0. Then the set of the edges of T which carry > 0 coordinates of l is isotopic to the lamination l.
Proof. — Follows immediately from the parts b) and c) of Lemma 14.2.
The part b) of the theorem is an immediate consequence of Corollary 14.1. The theorem is proved.
14.2. Corollary.
a) Any rational X-lamination without loops admits a non-negative coordinate system.
b) The set of rational X-laminations without loops is dense in the space XL(S;Q) of all rational X-laminations.
Proof. — a) It has been proved during the proof of Theorem 14.1.
b) The intersection of the domain described in Theorem 14.1 with the space of all rational X-laminations is described as the set of the ones without closed loops.
The corollary is proved.
So a little perturbation destroys loops in a lamination.
It is tempting to avoid the use of the Morse–Smale theory in the proof, making the proof self-contained. Here is perhaps a first step in this direction.
14.3. Lemma. — Let l be a real X-lamination and α be a closed leaf of it. Take any 3-valent graph and let now xi be the coordinates of l and ai be the coordinates of α considered as an A-lamination. Then
ixiai =0.
Proof. — The intersection pairing of α with l equals zero. On the other hand it is computed by the formula
ixiai. The lemma is proved.
This lemma implies that, since all ai’s are integers, if all xi’s are linearly inde-pendent over Q , there are no closed curves. However this does not prove yet that for a generic real X-lamination all curves of the lamination go between the punctures:
we have to take care of the non-closed curves winding inside of S.
4. An application to the canonical map IX. — It follows from the part a) of Corol-lary 14.2 that the restriction of the canonical mapIX to the integral laminations with-out loops is especially simple. Namely, given such a laminationl, in every non-negative coordinate system forl the elementIX(l)is given by a product of the coordinate func-tions in this coordinate system. So IX(l) is in the cluster algebra corresponding to the space ASL2,S. The image of laminations containing loops does not belong to it.