Completions of Teichmüller spaces and canonical bases

1. Cyclic sets, laminations and cacti-sets. — Let C be a cyclic set, perhaps infinite.

A simple lamination l in C is a disjoint collection, perhaps infinite, of pairs of elements (ci,c_i) of C so that for any i = j the order of the quadruple (ci,cj,c_i,c_j) is not com-patible with the cyclic order of C. (If C= S¹, this just means that the chords (ci,c_i) and(cj,c_j) do not intersect).

Gluing every pair of points ci,c_i we get a setC/l. It no longer has a cyclic struc-ture compatible with the projection π : C → C/l. A subset M of C/l is cyclic if it has a cyclic structure such that the projection π⁻¹M → M is a map of cyclic sets.

So a cyclic subset has a canonical cyclic structure inherited from C. Cyclic subsets are ordered by inclusion. The maximal elements are called the maximal cyclic subsets

of C/l. The set C/l is a union of its maximal cyclic subsets; any two of them are either disjoint, or have a single element intersection. We call such sets cacti-sets.

F^IG.13.1. — A cacti-setC/l.

Example. — Let C be a circle. A simple lamination l in C is given by the end-points of a collection of disjoint chords inside the circle. Shrinking all chords to end-points we get a cacti-set C/l – see Figure 13.1. Its circles are the maximal cyclic subsets.

Let S be a surface, with or without boundary. A simple lamination on S is a fi-nite collection of simple, non-contractible, mutually non-isotopic, and non-isotopic to boundary components, disjoint loops onS, considered modulo isotopy.

A simple lamination l on S gives rise to a simple lamination in the cyclic set G∞(S). Indeed, letl be the preimage ofl in the universal coverSofS. The endpoints ofl provide a lamination in the cyclic set G∞(S). Denote by G∞(S,l)the correspond-ing cacti-set. The connected components ofS−l are in bijection with maximal cyclic subsets of G∞(S,l).

2. The classical case. — Let M⁰,X be the Knudsen–Deligne–Mumford moduli space of configurations of points onCP¹parametrised by a finite set X. An embedding of sets X⊂ X gives rise to a canonical projection M⁰,X −→ M⁰,X – forgetting the points parametrized by X−X.

Let C be an infinite countable set. Set M0,C:=lim

←−M0,X

where the projective limit is over finite subsets X of C, and the maps correspond to inclusions of finite sets. (If C is a finite set, we recover M0,C).

Now let C be a cyclic set. ThenM0,C is equipped with a positive atlas, so there is its real positive part M0⁺,C. Its closure in M⁰,C(R) is denotedM⁺0,C. The set M⁺0,C

has a decomposition (into “cells” of a priori infinite dimensions) parametrized by lam-inations in C.

Example. — When C is a finite cyclic set, M⁺0,C is (the closure of ) the Stasheff polytope, with its natural cell decomposition parametrised by the laminations in C ([GM]).

The points of the Teichmüller space XPGL⁺ ₂,S are given by positive π1(S) -equi-variant maps G∞(S) → P¹(R). Thus they are points of the moduli space M0⁺,C for C=G∞(S).

13.1. Definition. — The set X⁺PGL2,S is the closure of XPGL⁺ 2,S in M0,C for C = G∞(S).

We need the following general definition. Given a simple lamination l on S, the boundary components of the curve S−l correspond either to the loops of the lam-ination l, or to the boundary components of S. The former are called l-boundary components of S−l.

13.2. Definition. — Letl be a simple lamination onS. The moduli spaceX_G,S−l^un paramet-rizes framedG-local systems onS−l with unipotent monodromies around the l-boundary components of S−l. The moduli space AG^un,S−l parametrizes twisted unipotent G-local systems on S−l with decorations at those boundary components which are inherited from S.

So XG^un,S−l is the preimage of the identity element under the projection X^G,S−l

−→ H^k, given by the semi-simple parts of the monodromies around the l-boundary components onS−l. The moduli spaceAG^un,S−l is the quotient ofAG,S−l by the action of the group H^k provided by the l-boundary components on S−l. Therefore each of them carries an induced positive atlas, and thus there are the real positive parts X_G,S−l^+,un and A_G,S−l^+,un.

13.1. Proposition. — X⁺PGL₂,S is a disjoint union of cells parametrised by simple lamina-tions in S:

X⁺PGL₂,S=

-l

XPGL^+,un2,S−l; where l are simple laminations on S.

An embedding of the right hand side to the left for any Gis defined in Section 13.3.

Apparently the action of the mapping class group ΓS extends to the closure X⁺PGL₂,S. The quotient U⁺PGL₂,S/ΓS is identified with the Knudsen–Deligne–Mumford moduli spaceM^S, usually denotedM^g,n, where g is the genus of Sand n is the num-ber of punctures onS.

3. Completions of higher Teichmüller spaces. — Recall that a positive configuration of flags parametrised by a cyclic set C is the same thing as a positive map

C→B(R) modulo G(R)-conjugation. Recall the canonical projection π :G∞(S)→ G∞(S,l).

13.3. Definition. — Let C and C be cyclic sets related by a surjective map of cyclic sets π:C→C.

A sequence {ψn} of positive configurations of flags parametrised byCis convergent to a positive configuration ψ parametrised by C if there are maps ψn :C→B(R) representing configurations ψⁿ such that the limit ψ:=limn→∞ψⁿ exists, factors through C, i.e. ψ(ci)=ψ(c_i)if π(ci)= π(c_i), and the induced map ψ:C →B(R) represents the configuration ψ.

13.4. Definition. — A sequence of points ψⁿ ∈XG⁺,S is convergent if there exists a simple lamination l on S such that for every maximal cyclic subset µ of G∞(S,l) the sequence of positive configurations

ψn^µ :π⁻¹(µ)−→B(R), (13.1)

obtained by restrictions of ψⁿ’s to π⁻¹(µ), is convergent to a positive configuration ψ^µ : µ −→

B(R).

Since the maps (13.1) correspond to points of XG⁺,S, they areπ1(S)-equivariant.

Therefore the limiting map ψ^µ is π1(S_µ)-equivariant, where S_µ is the component of S−l corresponding to µ.

Recall that there exists a unique regular unipotent conjugacy class in G(R): it is given by the maximal Jordan block for G=PGLn.

13.1. Lemma. — The monodromy of the local system on S_µ corresponding to ψ^µ around an l-boundary component is regular unipotent.

Proof. — The monodromyM around a boundary component is conjugate to an element of B(R>0). Since the canonical flags corresponding to M and M⁻¹ coincide, M is unipotent. Any element of U(R>0) is regular. The lemma is proved.

Lemma 13.1 implies that a convergent sequence of points ψn ∈ XG⁺,S corres-ponds to a point of the space XG^+,,S^un−l, for a certain simple lamination l on S, which we view as its limit. The gluing from Section 7 shows that we can get any point of XG^+,un,S−l. Similar questions for the A-spaces are reduced to the ones for the X-spaces.

So we arrive at the following definition.

13.5. Definition. — Each of the sets B(XG⁺,S)andB(AG⁺,S)is a disjoint unions of cells, which are parametrised by simple laminations l in S:

B

XG⁺,S =

-l

XG^+,un,S−l; B

AG⁺,S =

-l

AG^+,un,S−l, wherel are simple laminations on S.

The topology on these sets is provided by Definition 13.4. The obtained topological space are called completions of the higher Teichmüller spacesX_G,S⁺ and A_G,S⁺ .

We use here a different notation, comparing to Definition 13.1, to emphasize that for G= PGL2 we do not know whether the space B(XG⁺,S)is the closure of the space XG⁺,S in any sense.

Here is how one should define a full completion of the higher Teichmüller space for a group G different from PSL2. One should be able to define a natural compact-ification Confn(B) of the space of configurations of n flags in generic position, so that forG=PSL2we recover the moduli space M0,n. Then we would define a completion of the space LG⁺,S as the closure Conf⁺_{G∞(S),π1(S)}(B) of the positive π1(S)-equivariant configuration space Conf⁺_{G∞(S),π1(S)}(B) in the above compactification.

4. The canonical map for closed surfaces, and its behavior at the boundary. — For a closed surface S the moduli space AG,S is isomorphic to the moduli space of G-local systems on S, although this isomorphism is non-canonical if sG = e. So it has a canonical symplectic structure. WhenG has trivial center, the set XG,S(Z^t)was defined in Section 6.9.

13.1. Conjecture. — Let Sbe a surface without boundary. Assume thatGhas trivial center.

Then there exists a canonical map

I:X^G,S(Z^t)−→O(A^LG,S).

(13.2)

Its image is a basis of O(A^LG,S). The map (13.2) has a quantum deformation.

We conjecture that the canonical map (13.2) behaves nicely at the boundary of the moduli spaceA^LG,S, and this requirement determines it uniquely. Let us formulate this precisely.

Recall the embedding

βl :A_G,S−l^+,un _→BAG⁺_{,S .} (13.3)

Let Ol(AG,S) be the subspace of functions which have limit at the component (13.3).

So by the very definition there is a restriction map Rl :Ol(A^LG,S)−→O

A^LunG,S−l . (13.4)

The space on the right should carry a canonical basis: the Duality Conjecture for the surface S−l implies that there should be a canonical map

XG^un,S−l(Z^t)−→O

A^LunG,S−l . (13.5)

Indeed, the canonical map I_X :XG,S−l(Z^t)−→O(A^L_G,S−l)should have the following property: if x ∈ XG,S−l(Z^t), and y ∈ H^k(Z^t) is its image under the map provided by the projection XG,S−l → H^k, then the function I_X(x) transforms under the action of the dual torus ^LH^k by the character χy corresponding to y. Thus the functions cor-responding to points of the subset X_G,S−l^un (Z^t) should be invariant under the action of^LH^k. So the map I_X should restrict to the map (13.5).

Let V be a vector space with a given basis B, V0 a subspace of V, and r:V0→W a surjective linear map. We say that the basis B restricts under the map r to a basis in W if:

(i) The basis elements which lie inside of V0 generate V0, i.e. provide a basis B0 in V0.

(ii) Let B₀ be the subset of the basisB0 consisting of the elements which are not killed by r. Then r(B₀) is a basis in W.

13.2. Conjecture. — Let Sbe a surface without boundary. Then, for any simple lamination l on S, the restriction map Rl is surjective, and the canonical basis (13.2) in O(A^LG,S) restricts under the map Rl to the canonical basis in O(A^Lun_G,S−l).

In other words, for any simple laminationl on S, we should have a commutative diagram

XG^⊥,^lS(Z^t) −−−→ Ol(A^LG,S)

⏐⏐ ⏐⏐^Rl XG^un,S−l(Z^t) −−−→ O

A^LunG,S−l . (13.6)

Here the bottom arrow is the map (13.5). The top arrow is given by the restriction of the map I.

The canonical maps for closed surfaces should be uniquely determined by Con-jecture 13.2 and the canonical maps for surfaces with boundary.

Example. — Conjectures 13.1 and 13.2 are valid for G=PGL2. Indeed, in this case the set XG,S(Z^t) coincides with the set of integral laminations on S, i.e. collec-tions of simple disjoint curves with positive integral weights on S, considered modulo isotopy. The canonical map assigns to a lamination

ini{αⁱ} the function

iTrMⁿ_αⁱ_i on the moduli space A^SL2,S of twisted SL2-local systems on S. Here αⁱ is the lift of the loop αⁱ to the punctured tangent bundle of S. The quantum version of this map is nothing else but the Turaev algebra of S. The subset XG^⊥,^lS(Z^t) consist of the lam-inations with the zero intersection index with l. The left vertical map in (13.6) kills the lamination l. So in this case the diagram (13.6) is well-defined and commuta-tive.