Tau-functions \`a la Dub\'edat and probabilities of cylindrical events for double-dimers and CLE(4)

Building upon recent results of Dub\'edat (see arXiv:1403.6076) on the convergence of topological correlators in the double-dimer model considered on Temperleyan approximations $\Omega^\delta$ to a simply connected domain $\Omega\subset\mathbb C$ we prove the convergence of probabilities of cylindrical events for the \emph{double-dimer loop ensembles} on $\Omega^\delta$ as $\delta\to 0$. More precisely, let $\lambda_1,\dots,\lambda_n\in\Omega$ and $L$ be a macroscopic lamination on $\Omega\setminus\{\lambda_1,\dots,\lambda_n\}$, i.e., a collection of disjoint simple loops surrounding at least two punctures considered up to homotopies. We show that the probabilities $P_L^\delta$ that one obtains $L$ after withdrawing all loops surrounding no more than one puncture from a double-dimer loop ensemble on $\Omega^\delta$ converge to a conformally invariant limit $P_L$ as $\delta \to 0$, for each $L$. Though our primary motivation comes from 2D statistical mechanics and probability, the proofs are of a purely analytic nature. The key techniques are the analysis of entire functions on the representation variety $\mathrm{Hom}(\pi_1(\Omega\setminus\{\lambda_1,\dots,\lambda_n\})\to\mathrm{SL}_2(\mathbb C))$ and on its (non-smooth) subvariety of locally unipotent representations. In particular, we do \emph{not} use any RSW-type arguments for double-dimers. The limits $P_L$ of the probabilities $P_L^\delta$ are defined as coefficients of the isomonodormic tau-function studied by Dub\'edat with respect to the Fock--Goncharov lamination basis on the representation variety. The fact that $P_L$ coincides with the probability to obtain $L$ from a sample of the nested CLE(4) in $\Omega$ requires a small additional input, namely a mild crossing estimate for this nested conformal loop ensemble.


Introduction and main results
Convergence of double-dimer interfaces and loop ensembles to SLE(4) and CLE(4), respectively, is a well-known prediction made by Kenyon after the introduction of SLE curves by Schramm, see [23,Section 2.3]. In particular, this provided a strong motivation to study couplings between Conformal Loop Ensembles (CLE) and the twodimensional Gaussian Free Field (GFF), a subject which remained very active during the last fifteen years and led to several breakthroughs in the understanding of SLEs and CLEs via the Imaginary Geometry techniques, e.g. see [17] and references therein.
Originally, this prediction was strongly supported by the convergence of dimer height functions to the GFF proved (for Temperleyan approximations on Z 2 ) by Kenyon [12,13] and the fact that the level lines of the GFF are SLE(4) curves, see [24] and [29]. More recently it received even more support due to the breakthrough works of Kenyon [14] and Dubédat [7] on the convergence of topological observables for doubledimer loop ensembles. Our paper should be considered as a complement to the work of Dubédat who writes (see [7,Corollary 3]) "By general principles, . . . the assumptions 1. (µ δ ) δ is tight, and 2. a probability measure µ on loop ensembles in a simply-connected domain D is uniquely characterized by the expectations of the functionals . . . imply weak convergence of the µ δ 's to the CLE 4 (D) measure as δ → 0." To the best of our knowledge, there are still no available results on the first assumption (tightness), thus Kenyon's prediction should not be considered as fully proven yet. The main goal of this paper is to give a solid ground to the second assumption: we show that the topological observables treated by Dubédat in [7] do characterize the measure on loop ensembles in the sense which is described below.
It is worth noting that several approaches to the convergence of (double-)dimer height functions to the GFF are known nowadays (e.g., see [2] and [4]) besides the original one of Kenyon [12,13], which is based on the analysis of the scaling limit of the Kasteleyn matrix by means of discrete complex analysis and is also the starting point for [14] and [7]. Also, the choice of discrete approximations Ω δ to a simply connected domain Ω is a very delicate question, see [22] for another (not Temperleyan) special case when the discrete complex analysis machinery works well. To be able to build upon the results of [7], below we assume that Ω δ are Temperleyan approximations on the square grids of mesh δ though this setup can be enlarged in several directions.
Recall that, given a Temperlean simply connected discrete domain Ω δ ⊂ δZ 2 , a double-dimer loop ensemble on Ω δ is obtained by superimposing two dimer configurations on Ω δ chosen independently uniformly at random: this produces a number of loops and double-edges, the latter should be withdrawn. We denote by Ξ δ Ω the random collection of simple pairwise disjoint loops obtained in this way. The nested conformal loop ensemble CLE (4) in Ω is a conjectural limit of Ξ δ Ω as δ → 0. The CLE(4) can be defined and effectively studied purely in continuum, see [25,20] and references therein for background. We denote by Ξ Ω a random sample of this loop ensemble. Note that Ξ Ω almost surely contains infinitely many loops but most of them are very small: almost surely, for each cut-off > 0 only finitely many of the loops of Ξ Ω have diameter greater than ε.
Let λ 1 , . . . , λ n ∈ Ω be a collection of pairwise distinct punctures in Ω. A lamination Γ is a finite collection of disjoint simple loops in Ω \ {λ 1 , . . . , λ n } considered up to homotopies. We call a lamination macroscopic if each of these loops surrounds at least two of the punctures. For a random loop ensemble Ξ and a deterministic macroscopic lamination Γ, let Ξ ∼ Γ denote the event that withdrawing all loops surrounding no more than one puncture from Ξ one obtains Γ. We call the events Ξ ∼ Γ cylindrical : their probabilities (for all n ≥ 1, all λ 1 , . . . , λ n ∈ Ω and all macroscopic laminations L) determine the law of Ξ for reasonable topologies on the space of loop ensembles.
Given n, put n e points at the interior of each edge e ∈ E.
An important notion that is constantly used in our paper is the complexity |Γ| of a lamination Γ, let us emphasize that this is not the number of loops in Γ, which we denote by #loops(Γ). Given λ 1 , . . . , λ n we fix a triangulation T Ω of Ω \ {λ 1 , . . . , λ n } and define |Γ| to be the minimal number of intersections of loops constituting Γ with the edges e ∈ E of T Ω . In fact, one can parameterize laminations on Ω \ {λ 1 , . . . , λ n } by multi-indices n = (n e ) ∈ Z E ≥0 satisfying certain conditions, see Fig. 1 and Section 3.2 for more details. Under this parametrization one has |Γ| = e∈E n e .
We denote by X := Hom(π 1 (Ω \ {λ 1 , . . . , λ n }) → SL 2 (C)) the (smooth) variety of SL 2 (C)-representations of the free non-abelian fundamental group π 1 (Ω\{λ 1 , . . . , λ n }). Note that one could view X as SL 2 (C) n by fixing generators of the fundamental group but this viewpoint is not invariant enough for the analytic tools that we use below. Let (λ i ) denote the loop surrounding a single puncture λ i and X unip := {ρ ∈ X : Tr (ρ((λ i ))) = 2 for all i = 1, . . . , n} be the (non-smooth) subvariety of locally unipotent representations ρ ∈ X. In this paper we study entire functions f : X → C or f : X unip → C, in the latter case we mean that f is continuous on X unip and is holomorphic on its regular part. Moreover, we are interested only in those entire functions that are invariant under the action of SL 2 (C) on X or X unip given by the conjugation ρ(·) → A −1 ρ(·)A, A ∈ SL 2 (C). Below we use the notation Fun hol (X) SL 2 (C) and Fun hol (X unip ) SL 2 (C) for these classes. Given a lamination (not necessarily macroscopic) Γ on Ω \ {λ 1 , . . . , λ n } we set Tr (ρ(γ)), ρ ∈ X. (1.1) Since Tr A = Tr A −1 for A ∈ SL 2 (C), this definition does not require to fix an orientation of the loops γ ∈ Γ. Clearly, one has f Γ ∈ Fun hol (X) SL 2 (C) and this function can be also treated as an element of Fun hol (X unip ) SL 2 (C) by taking the restriction to locally unipotent monodromies ρ ∈ X unip . The main results of our paper can be loosely formulated as follows: each invariant entire function f ∈ Fun hol (X) SL 2 (C) admits a unique expansion via the functions f Γ and each function f ∈ Fun hol (X unip ) SL 2 (C) admits a unique expansion via the functions f Γ indexed by macroscopic laminations Γ, with coefficients decaying faster than exponentially.
For ρ ∈ X unip , let  Following [7] we call the functions τ δ and τ topological correlators of the loop ensembles Ξ δ and Ξ , respectively. While τ δ is actually a finite linear combination of f Γ , one should be more careful with the infinite series (1.3). It is checked in [7] that P dbl-d [ Ξ Ω ∼ Γ ] = O(R −|Γ| 0 ) for some R 0 > 1, therefore τ is correctly defined at least in a vicinity of the trivial representation Id ∈ X unip . It seems to be known in the folklore that these probabilities actually decay super-exponentially as |Γ| → ∞ (e.g., see [30,Section 4] and references therein for related results) but we were unable to find an explicit reference to this fact and thus prefer to keep it as an assumption in Corollary 1.5, see also Remark 1.3.
Although one usually considers an isomonodromic tau-function as a function of λ 1 , . . . , λ n , the one discussed above can be also viewed as a function of a locally unipotent representation ρ as its multiplicative normalization does not depend on ρ. In our paper both Ω and λ 1 , . . . , λ n can be usually thought of as fixed once forever, thus from now onwards we use the shorthand notation τ (ρ) := τ Ω (λ 1 , . . . , λ n ; ρ), ρ ∈ X unip . (1.4) From the above construction one can see that τ ∈ Fun hol (X unip ) SL 2 (C) , note that this also can be easily deduced from the following theorem, which is the main result of [7].
Theorem 1.1 (Dubédat). Let Ω be a planar simply connected domain, Ω δ be a sequence of Temperleyan approximations to Ω, and λ 1 , . . . , λ n ∈ Ω. Then, the following holds: (i) For each locally unipotent representation ρ ∈ X unip one has and the convergence is uniform on compact subsets of X unip .
The next theorem is the main result of our paper.
Each entire function f ∈ Fun hol (X unip ) SL 2 (C) admits a unique expansion where the functions f Γ are given by (1.1) and |Γ| −1 log |p Γ | → −∞ as |Γ| → ∞. Moreover, for each R > 0 there exists a compact subset K R ⊂ X unip and a constant C R > 0 independent of f such that one has for all macroscopic laminations L.
Remark 1.3. In fact, we prove a slightly stronger result as we require f to be defined only on a bounded subset of X unip , see Theorem 5.11 for the precise statement. It is worth noting that our results do not guarantee the uniqueness of the expansion (1.5) for functions defined just in a small vicinity of Id ∈ X unip . To illustrate a possible catch one can think about expanding entire functions of one complex variable in the basis 1, z − 1, z 2 − z, . . . , z n − z n−1 , . . . . In the full plane such expansions always exist and are unique but 1 + (z − 1) + (z 2 − z) + · · · = 0 in a vicinity of the origin. Since the functions f Γ are far from being a Fourier basis we expect a similar (though more involved) phenomenon in our setup.
It is easy to see that a combination of Theorem 1.1 and Theorem 1.2 imply the convergence of probabilities of cylindrical events. Let be the expansion of the isomonodromic tau-function provided by Theorem 1.2.
Let Ω be a planar simply connected domain, Ω δ be a sequence of Temperleyan approximations to Ω, and λ 1 , . . . , λ n ∈ Ω. Then, for each macroscopic lamination Γ on Ω \ {λ 1 , . . . , λ n }, one has Moreover, R |Γ| · |p δ Γ − p iso Γ | → 0 as δ → 0 uniformly in Γ for each R > 0. Proof. By definition, p δ Γ are nothing but the coefficients in the expansion (1.5) of the function τ δ . Therefore, for each R > 0, Theorem 1.2 implies the uniform estimate and the right-hand side vanishes as δ → 0 due to Theorem 1.1(i). Corollary 1.5. In the same setup, assume that (1.7) Proof. If the assumption (1.7) holds, then the series (1.3) converges for all ρ ∈ X unip and τ ∈ Fun hol (X unip ) SL 2 (C) . Due to Theorem 1.1(ii) the entire functions τ and τ coincide in a vicinity of ρ = Id and hence everywhere on X unip . Therefore, the uniqueness of the expansion (1.5) gives p iso Γ = p Γ for all macroscopic laminations Γ. The main ideas of the proof of Theorem 1.2 are discussed in Section 2. We conclude the introduction by the following vague remark on a potential deformation of the topological observables (1.3). Though the first naive idea would be just to replace the CLE(4) measure in their definition by CLE(κ) with κ = 4, thus obtained functions do not look very natural. In view of the material discussed in Section 4 it actually looks more promising to simultaneously deform the functions f Γ given by (1.1) by using the quantum trace functionals [3] instead of the usual traces. Having in mind the famous predictions on the scaling limits of loop O(n) and FK(q) models (e.g., see [23,Section 2.4] and [26, Section 2]), it sounds plausible that the quantization parameter should be then tuned so that the simple loop weight in the corresponding skein algebras is equal to ±2 cos(4π/k). We believe that developing tools to analyze such deformations might be of great interest, both from CLE and lattice models perspectives.
2. Toy example n = 2 and the strategy of the proof In this section we informally describe the main ideas of the proof of Theorem 1.2. Sometimes we use the case n = 2 as a toy example. Certainly, this is a classical and well-studied setup (e.g., see [14, Section 10] and [7, Corollary 2]): each macroscopic lamination on Ω \ {λ 1 , λ 2 } is given by k ≥ 0 loops homotopic to the simple loop l surrounding both punctures. Let z := Tr (ρ(l)) ∈ C and p δ k be the probability to see exactly k copies of l in the double-dimer model loop ensemble on Ω δ , divided by 2 k . In this situation our main result can be rephrased as follows: the convergence of entire functions τ δ (z) = k≥0 p δ k z k → τ (z) = k≥0 p k z k as δ → 0 on compact subsets of C implies that p δ k → p k as δ → 0 for each k ≥ 0. Unfortunately, such a straightforward argument does not work for n > 2 since the set of functions f Γ : X unip → C indexed by macroscopic laminations does not have a structure of the Fourier basis. Though one can relatively easily deduce from the Fock-Goncharov theorem [8,Theorem 12.3] that f Γ form an algebraic basis of the space Fun alg (X unip ) SL 2 (C) , even the uniqueness of expansions f (ρ) = Γ−macroscopic p Γ f Γ (ρ) of entire functions f ∈ Fun hol (X unip ) SL 2 (C) is not at all obvious: to see a possible difficulty, the reader can think about replacing the Fourier basis 1, z, z 2 , . . . by its lower-diagonal transform 1, z − 2, z 2 − 4z, . . . , z n − 2 n z n−1 , . . . in the toy example given above.
Even if the aforementioned uniqueness issue is settled, it still might be problematic to extract the convergence of coefficients p δ k from the convergence of functions τ δ unless an a priori estimate similar to (1.6) is available, cf. Remark 1.3. Since, to the best of our knowledge, no explicit analogue of the Cauchy formula (which settles the toy case n = 2) on X unip is known if n > 2, we develop a set of general tools to analyze SL 2 (C)-invariant entire functions on X unip as sketched below.
Recall that the manifold X = Hom(π 1 (Ω\{λ 1 , . . . , λ n }) → SL 2 (C)) can be viewed as SL 2 (C) n (a parametrization is given by a choice of generators of the fundamental group) and hence it carries the structure of an algebraic group whereas its subvariety X unip does not. Therefore, it is easier to analyze the ring of invariant functions on X first. Following Fock and Goncharov [8,Section 12] this goes as follows: -One encodes the laminations Γ by multi-indices n = (n e ) e∈E ∈ Z E ≥0 indexed by the edges of a triangulation T Ω of Ω \ {λ 1 , . . . , λ n }, see Fig. 1. If n = 2, these multiindices are just triples (n 1 , n 2 , n 3 ) of non-negative integers (see Fig. 2) satisfying the so-called lamination condition n 1 + n 2 + n 3 is even, |n 1 − n 2 | ≤ n 3 ≤ n 1 + n 2 .
(2.1) -Flat SL 2 (C) connections on Ω \ {λ 1 , . . . , λ n } can be parameterized by putting transition matrices A e ∈ SL 2 (C) onto the edges of T Ω and factorizing over the natural action (change of the bases of the rank-two vector bundle) of SL 2 (C) matrices assigned to the faces of T Ω . This parametrization provides a natural isomorphism Fun alg (X) SL 2 (C) ∼ = Fun alg (SL 2 (C) E ) SL 2 (C) F .
-As already mentioned, the functions f (n 1 ,n 2 ,n 3 ) ∈ Fun alg (X) SL 2 (C) (for simplicity, we focus on the case n = 2 though the same arguments work well in the general case) do not have a Fourier basis structure. Nevertheless, there exists another collection of functions g (n 1 ,n 2 ,n 3 ) ∈ Fun alg (X) SL 2 (C) indexed by the same set of triples satisfying the lamination condition (2.1) which is an orthogonal basis in the space L 2 (SU 2 (C) E ) SU 2 (C) F , we call these functions the Peter-Weyl basis. In fact, they can be constructed by the following symmetrization procedure. Let C(n 1 , n 2 , n 3 ) be the set of all possible collections of (not necessarily simple or disjoint) loops obtained by concatenating two collections of arcs with (n 1 , n 2 , n 3 ) endpoints drawn inside each of the two faces of T Ω so that no arc connects two points on the same edge, see where f C (ρ) := γ∈C Tr (ρ(γ)).
-The Fock-Goncharov theorem claims that the bases f Γ and g Γ are related by a lower-triangular transform: where the partial order on the set of laminations is given by the partial order on the set of multi-indices n ∈ Z E ≥0 . By analogy with entire functions one can expect that each holomorphic function f ∈ Fun hol (X) SL 2 (C) admits a Fourier-type expansion f = Γ q Γ g Γ with coefficients q Γ decaying faster than exponentially. Formally, this implies that one can also but there is a catch: to pass from the former series to the latter rigorously one needs an exponential upper bound for the coefficients c Γ∆ of the Fock-Goncharov change of the bases. To the best of our knowledge, such an estimate is not available in the existing literature thus we prove the (non-optimal) upper bound |c Γ∆ | ≤ 4 |Γ| in our paper. It is well-known that the identities between the functions f Γ , f C and g Γ can be equivalently written in terms of the Kauffman skein algebras Sk(M, ±1) of framed knots in M := (Ω \ {λ 1 , . . . , λ n }) × [0, 1]. In fact, Fun alg (X) SL 2 (C) ∼ = Sk(M, −1) ∼ = Sk(M, 1) and the skein relations reflect the identity Tr (AB) + Tr (AB −1 ) = Tr (A)Tr (B) for A, B ∈ SL 2 (C). This isomorphism suggests the idea of expanding each of the terms f C in the definition of g Γ as f C = ∆:∆≤Γ c C∆ f ∆ by resolving all the crossings of C via the skein relations. Unfortunately, this is not an easy thing to do: a collection of loops C may contain about |Γ| 2 crossings, hence one must analyse highly non-trivial cancelations arising along the way in order to end up with an exponential bound for c C∆ .
Following the advice of Vladimir Fock we circumvent these complications by using the positivity of the co-called bracelet basis of Sk(M, 1) proved by D. Thurston [28] in combination with a representation of Sk(M, 1) in the space of Laurent polynomials coming from Thurston's shear coordinates of hyperbolic structures on Ω \ {λ 1 , . . . , λ n } (e.g., see [3]). Remarkably enough, in this representation all functions f C are mapped to Laurent polynomials with positive integer coefficients. Together, these positivity results imply the desired exponential estimate of coefficients as the sums contain no cancellations anymore.
The arguments briefly described above allows one to prove counterparts of our main results for holomorphic functions living on the whole manifold X and their expansions in the basis f Γ indexed by all, not necessarily macroscopic, laminations. The last but not the least part of our analysis is devoted to the translation of these results to holomorphic functions living on the subvariety X unip ⊂ X. Let (λ i ) denote the loop surrounding a single puncture λ i . Note that each lamination admits a unique decomposition into a macroscopic lamination Γ, k 1 copies of (λ 1 ), k 2 copies of (λ 2 ) etc., we use the notation Γ (λ k ) to describe such a lamination.
The main ingredient of the proof of the existence part of Theorem 1.2 is a 'controlled' extension of holomorphic functions from X unip to X provided by Manivel's Osawa-Takegoshi-type theorem [15]. Roughly speaking, given f ∈ Fun hol (X unip ) SL 2 (C) one can first extend it to a function F ∈ Fun hol (X) SL 2 (C) and then group all the terms of the expansion of F that correspond to the laminations Γ (λ k ) with a fixed Γ: Finally, the uniqueness part of Theorem 1.2 can be deduced from a version of Hilbert's Nullstellensatz for invariant holomorphic functions on X as follows. Assume that a series f := Γ−macroscopic p Γ f Γ vanishes on X unip . Since X unip is cut off from X by the equations f (λ i ) − 2 = 0, one can find functions h i ∈ Fun hol (X) SL 2 (C) such that Due to the existence of expansions (1.5) one has . Using the uniqueness of the expansion of f ∈ Fun hol (X) SL 2 (C) in the basis f Γ (λ k ) and the fact that the product f Γ (λ k ) · f (λ i ) is again a basis function one obtains the identity by collecting all the terms corresponding to a given macroscopic lamination Γ. The right-hand side vanishes at ρ = Id and hence p Γ = 0.
Certainly, the informal discussion given in this section is far from being complete or rigorous, with many important details omitted. For instance, we actually work with functions F ∈ Fun hol (B R ) SU 2 (C) F defined on poly-balls B R ⊂ (C 2×2 ) E rather than with F ∈ Fun hol (SL 2 (C) E ) SL 2 (C) F as sketched above. Nevertheless, we hope that this discussion might help the reader to understand the general structure of our arguments and the set of tools used in the proof.
The rest of the paper is organized as follows. We collect relevant basic facts of the representation theory of SL 2 (C) and discuss the Fock-Goncharov theorem in Section 3. Section 4 is devoted to the proof of the exponential estimate of the coefficients c Γ∆ : as explained above, the Kauffman skein algebra Sk(M, 1) plays a central role here. In Section 5 we prove our main results. In particular, Section 5.2 is devoted to holomorphic extensions of functions defined on compact subsets of X unip and Section 5.3 contains a precise version of the Nullstellensatz that we need. All these ingredients are used in Section 5.4 to prove Theorem 5.11 which is a slightly stronger version of Theorem 1.2.

Preliminaries and Fock-Goncharov theorem
Let X = Hom(π 1 (Ω\{λ 1 , . . . , λ n }) → SL 2 (C)) be the affine variety parameterizing representations of π 1 (Ω \ {λ 1 , . . . , λ n }) in SL 2 (C), note that X can be also thought of as an algebraic variety. Let Fun alg (X) be the ring of algebraic functions on X. The group SL 2 (C) acts on X by conjugations and, since this action is algebraic, one can consider the ring of invariants Fun alg (X) SL 2 (C) . Clearly, all functions f Γ belong to Fun alg (X) SL 2 (C) . A famous theorem due to Fock and Goncharov [8,Theorem 12.3] states that these functions actually form a basis in the vector space Fun alg (X) SL 2 (C) . For the sake of completeness and in order to introduce a consistent notation, we begin this section with some basic facts of the representation theory of SL 2 (C) and then repeat the proof of this theorem in Section 3.4 following [8]. In Section 3.5 we discuss extensions of the functions f Γ and g Γ from SL 2 (C) E to the Euclidean space (C 2×2 ) E . Finally, Section 3.6 is devoted to the lower bound for the norms of thus obtained extensions of g Γ which play an important role in the core part of the paper.
3.1. Basics of the representation theory of SL 2 (C). In this section we collect basic facts of the representation theory of SL 2 (C). We use the well known correspondence between representations of groups and their Lie algebras, which holds for SL 2 (C). Due to this correspondence, one can work with the Lie algebra sl 2 (C) instead of SL 2 (C) itself. The proofs are mostly omitted, an interested reader can easily find them in the classical literature (e.g., see [9, Lecture 11] for a nice exposition).
Lemma 3.1. The Lie algebra sl 2 (C) of SL 2 (C) is given by Consider now an irreducible finite-dimensional representation sl 2 (C) → End(V ).
(ii) There exists a non-zero vector v ∈ V such that the following holds: v is an eigenvector for h; The above considerations can be summarized as follows: The irreducible finite-dimensional representations of sl 2 (C) are enumerated by non-negative integers and dim V n = n + 1, where V n denotes the n-th representation. For each n, the representation V n has a basis v 0 , v 1 , . . . , v n such that (ii) The representation V 1 corresponds to the standard matrix representation of SL 2 (C) and V n is isomorphic (as sl 2 (C) module) to the space Sym n V 1 , which is in its turn isomorphic to the space of homogeneous polynomials of two variables of degree n. Under Recall that the action of sl 2 (C) on the tensor product of representations is defined as a → a ⊗ Id + Id ⊗ a.
Lemma 3.4. (i) For each n, m ≥ 0, the following isomorphism of sl 2 (C) modules holds: The projection onto the first component is given by the symmetrization if n−m is even and l(n, m) = 0 if n−m is odd. The projection onto the first component is given by the symmetrization V ⊗n Given a representation SL 2 (C) → End(V ), we denote by V SL 2 (C) the subspace of invariant (under the action of SL 2 (C)) vectors in V .
Otherwise, this invariant subspace is trivial.
Proof. Using (3.1) one gets appear only once in the right-hand side of (3.1). The claim follows.
Following [8], one can construct a bijection between L(T Ω ) and the set of laminations as follows. Given n ∈ L(T Ω ) and an edge e ∈ E, let Γ e be an arbitrary collection of n e distinct points lying on the interior of e. Consider a face σ ∈ F with ∂σ = {e 1 , e 2 , e 3 }. The condition (3.3) for (n e 1 , n e 2 , n e 3 ) ensures that we can draw 1 2 (n e 1 + n e 2 + n e 3 ) non-intersecting chords inside σ, each of them starting at some point from Γ e i and ending at some point from Γ e j with i = j. Moreover, such a drawing is unique up to homotopies. Once all such chords in all faces σ are drawn and concatenated in a natural way, one obtains a collection of simple disjoint curves that represents a lamination Γ = Γ(n), see Figure 1.
Conversely, given a lamination Γ there is a unique way to represent each of the curves γ ∈ Γ as a closed non-backtracking path on the dual graph T • Ω . Set n e to be the total number of crossing of an edge e by these paths on T • Ω . It is easy to see that n := (n e ) e∈E ∈ L(T Ω ) and that the procedure explained above leads to Γ(n) = Γ. Thus, we have constructed a bijection between the set of all laminations and the set L(T Ω ).
We also obtain a partial order: given two laminations Γ 1 , Γ 2 and the corresponding multi-indices n 1 , n 2 ∈ L(T Ω ) we say that We define the complexity |Γ| of a lamination Γ as the sum of all coordinates of the corresponding multi-index n, it is worth noting that this definition depends on the (arbitrary) choice of the triangulation T Ω of Ω. Finally, given a lamination Γ and a face σ we say that c is a chord of Γ in σ if c is one of the chords drawn inside σ along the procedure explained above, see Figure 1.

3.3.
Peter-Weyl basis. Following [8], in this section we discuss the application of the (algebraic) Peter-Weyl theorem to the space Fun alg (X) SL 2 (C) . For this purpose we move from SL 2 (C)-representations of π 1 (Ω \ {λ 1 , . . . , λ n }) to holonomies of flat SL 2 (C) connections on Ω\{λ 1 , . . . , λ n }. These connections can be parameterized by collections (A e ) e∈E of matrices assigned to edges e ∈ E: one can think that there is a copy C 2 σ of C 2 assigned to each of the faces σ of the triangulation and that A e : C 2 σ left → C 2 σ right encodes the change of the bases when moving across e from left to right (recall that we have fixed an orientation of all the edges e ∈ E once forever). This parametrization allows one to study the ring of functions Fun alg (X) as a subring of the ring Fun alg (SL 2 (C) E ). The action of SL 2 (C) on X corresponds to an action of the bigger group SL 2 (C) F on SL 2 (C) E , the latter is given by changes of bases in the spaces C 2 σ assigned to faces of T Ω . Namely, given where σ left and σ right stand for two faces adjacent to an edge e. Quotients under these actions of SL 2 (C) and SL 2 (C) E coincide (see Lemma 3.7 below) and so the rings of invariants do: (3.5) Then one can use the representation theory of the group SL 2 (C) E to analyze the ring of SL 2 (C) F -invariant functions on it. More details are given below.
Classically, given an algebraic group G and a representation ρ : The following algebraic version of the Peter-Weyl theorem asserts that any algebraic function on SL 2 (C) E can be obtained in this way.
Theorem 3.6 (algebraic Peter-Weyl theorem). Let G be a reductive linear algebraic group, Fun alg (G) denote the space of algebraic functions on G and G be the set of all irreducible finite-dimensional representations of G. Then the mapping (3.6) defines an isomorphism of G × G-modules: Proof. E.g., see [27,Theorem 27.3.9].
Given a path γ = (σ 0 σ 1 . . . σ m ) on the dual graph T • Ω and A ∈ SL 2 (C) E , let e 0 , e 1 , . . . , e m−1 ∈ E be the sequence of edges of T Ω crossed when one moves along γ and A e k be the matrices assigned to these edges. The holonomy of A along γ is defined as where sign(σ, e) = +1 if σ lies to the left of e and sign(σ, e) = −1 otherwise (recall that the orientation of all the edges e ∈ E is fixed once forever).
Proof. The fact that φ intertwines two actions is straightforward. Let T be a spanning tree for the triangulation T Ω and a subvariety Y ⊂ SL 2 (C) E be defined as It is easy to see that φ restricted to Y is an isomorphism, so let ψ : X → Y be the inverse mapping. Further, for each Thus the restriction mapping Fun alg (SL 2 (C) E ) → Fun alg (Y ) sends the subring of SL 2 (C) F -invariant functions on SL 2 (C) E isomorphically onto its image, which we denote by Fun alg ( provides an inverse mapping to φ * . Recall that we denote by V 1 the standard two-dimensional representation of SL 2 (C) and that all finite-dimensional irreducible representations of SL 2 (C) are provided by We now define two pairings on V n . To get the first we identify V 1 ∧ V 1 with C by setting x∧y = 1. This gives rise to a skew-symmetric bilinear form (v, w) → v∧w on V 1 . Once can extend it on V ⊗n 1 and then restrict to V n . The result is a non-degenerate (skew-symmetric if n is odd and symmetric if n is even) bilinear form on V n , which we still denote by v ∧ w.
The second pairing on V n is obtained in a similar manner from the Hermitian scalar product on V 1 (which is defined by x, y = 0 and x, x = y, y = 1 on V 1 ), extended to V ⊗n and then restricted to V n . One can easily see that We need an additional notation. For a face σ ∈ F and an edge e ∈ ∂σ adjacent to it, let V σ,e,n be a copy of V n . Given a multi-index n ∈ Z E ≥0 we introduce a space recall that σ left (e) is the face lying to the left of e and σ right (e) is the one to the right, with respect to the once forever fixed orientation of e. There is a linear mapping Λ E : V n → C given by . Define an action of the group SL 2 (C) E on V n as follows: One can now construct a linear mapping Applying Theorem 3.6 on each of the edges of E, one sees that (3.10) is an isomorphism. In order to study the subring of SL 2 (C) F -invariant functions on SL 2 (C) E , we rearrange the factors in the definition of V n and define an action of SL 2 (C) F on V n as Lemma 3.8. The mapping (3.10) commutes with the action of SL 2 (C) F .
,e,ne ∈ V n and Υ(v) be the corresponding algebraic function on SL 2 (C) E . Note that for each v, w ∈ V d and C ∈ SL 2 (C) one has Cv ∧ w = v ∧ C −1 w. Using this observation and (3.4) one gets thus the actions of SL 2 (C) F on SL 2 (C) E and V n commute with Υ.
One can now write and apply Corollary 3.5 to each of the SL 2 (C)-invariant subspaces corresponding to faces σ ∈ F. It follows that V if the lamination condition (3.3) holds true for all σ, and is trivial otherwise. Due to Lemma 3.7 and Lemma 3.8, this leads to the following decomposition of Fun alg (X) SL 2 (C) into a direct sum of one-dimensional spaces: in the decomposition (3.11), where n(Γ) ∈ L(T Ω ) is the multi-index corresponding to Γ as discussed in Section 3.2. We call functions g Γ the Peter-Weyl basis of Fun alg (X) SL 2 (C) . Remark 3.10. Since all the spaces in the decomposition (3.11) are one-dimensional, in order to prove that the functions g Γ indeed form a basis of Fun alg (X) SL 2 (C) , it is enough to check that these functions do not vanish. This fact follows from Lemma 3.13.
3.4. Lamination basis f Γ and Peter-Weyl basis g Γ via each other. A famous theorem due to Fock and Goncharov [8,Theorem 12.3] claims that the functions f Γ also form a basis in the space Fun alg (X) and that the corresponding change of bases between f Γ and g Γ is given by lower-triangular (with respect to the partial order on L(T Ω )) matrices: The main goal of this section is to set up a framework for the analysis of the coefficients c Γ∆ . While doing this we also repeat the proof of [8,Theorem 12.3].
Recall that a multi-curve is a smooth immersion c : m i=1 S 1 → Ω \ {λ 1 , . . . , λ n } of the union of a number of disjoint circles (m is not fixed), considered up to homotopies in the space of immersions: in Section 4 the total winding (rotation of the tangent vector) of multi-curves will be of importance. In particular, if some component of a multicurve has a nugatory self-crossing (i.e., a local twist that can be removed by the first Reidemeister move, see Fig. 3), then it may be not homotopic to a multi-curve having no such twists. We call a multi-curve minimal if it has neither nugatory self-crossings nor homotopically trivial components. For a lamination Γ, there is a unique minimal multi-curve homotopic to Γ. Of course, not all minimal multi-curves correspond to laminations: only those in which all intersections can be removed do.
Given a minimal multi-curve C one can view its components as non-backtracking loops on T • Ω and define n e to be the total number of intersections between an edge e and thus obtained loops on T • Ω . It is easy to see that n = (n e ) e∈E ∈ L(T Ω ) and hence there exists a lamination Γ(n). We denote Γ(C) := Γ(n) and |C| := |Γ(C)|.
Let γ 1 , ..., γ m be components of C. Define Our next goal is to study the image of f C under isomorphisms (3.11), in particular its image φ * (f C ) ∈ Fun alg (SL 2 (C) E ), see Lemma 3.7. Fix a multi-index n ∈ L(T Ω ) and let Γ be the lamination corresponding to n. Minimal multi-curves C with Γ(C) = Γ can be encoded by collections of permutations π ∈ σ∈F e∈∂σ Sym (n e ) in the following way. Let π σ,e ∈ Sym (n e ) be components of π and Γ e = {p e,1 , . . . , p e,ne } denote the set of intersection points of Γ and a given edge e ∈ E, enumerated from the beginning of e to its endpoint according to the fixed orientation of e. For each face σ ∈ F and for each pair of points p e 1 ,k 1 , p e 2 ,k 2 connected by a chord of Γ draw a simple smooth arc inside σ from p e 1 ,πσ,e 1 (k 1 ) to p e 2 ,πσ,e 2 (k 2 ) . Concatenating these arcs at the points p e,k in a smooth way one obtains a minimal multi-curve which we denote by C(Γ, π). We need even more notation. Given a multi-index n ∈ L(T Ω ), a face σ ∈ F, an edge e ∈ ∂σ adjacent to σ, and a number k such that 1 ≤ k ≤ n e , let W σ,e,k be a copy of V 1 . Introduce the space (3.12) Note that the space V n can be realized as a subspace of W n if one realizes V σ,e,ne as a result of the symmetrization in the space 1≤k≤ne W σ,e,k V ⊗ne 1 for each σ ∈ F and e ∈ ∂σ. One can extend the pairing Λ E and the action (3.9) of SL 2 (C) E from V n to W n by Λ E e∈E 1≤k≤ne w σ left (e),e,k ⊗ w σ right (e),e,k := e∈E 1≤k≤ne (w σ left (e),e,k ∧ w σ right (e),e,k ) and A e∈E 1≤k≤ne w σ left (e),e,k ⊗w σ right (e),e,k := e∈E 1≤k≤ne (w σ left (e),e,k ⊗A e w σ right (e),e,k ), respectively. Moreover, one can also lift the mapping Υ given by (3.10) from V n to W n using the same definition Υ(w) : We now choose an arbitrary orientation of the components of C(Γ, π) and set sign(C(Γ, π)) := (−1) #edges of T Ω crossed by C(Γ, π) from right to left, counted with multiplicity .
For a chord c of C(Γ, π) running from the k 1 -th point on e 1 to the k 2 -th point on e 2 inside σ, denote where v σ,e,k and v (1) σ,e,k correspond to the vectors x and y, respectively, under the identification of W σ,e,k and V 1 . Finally, denote w C(Γ,π) := sign(C(Γ, π)) · c -chord of C(Γ, π) u c,Γ,π ∈ W n . (3.13) It is easy to see that w C(Γ,π) does not depend on the choice of the orientation of C(Γ, π): changing the orientation of some of its components one changes the signs of all vectors u c,Γ,π corresponding to these components which is compensated by the same number of changes in the signs of crossings of C(Γ, π) and oriented edges e ∈ E.
On the other hand, expanding each of the vectors u c,Γ,π one obtains It is easy to see that the contribution of v provided that γ crosses e i from left to right. Otherwise, this contribution is given by . Therefore, all the signs cancel out and one gets . In the general case one simply repeats the same computation for each of the components γ 1 , . . . , γ m of a multi-curve C(Γ, π).
We now apply decomposition (3.2) for each of the edges e ∈ E and obtain Proof. The only non-trivial ingredient is to check that one has Υ(w) = 0 for w ∈ W 0 n . This immediately follows from the fact that Clearly, it is enough to prove this fact for elements v k ∈ V m , v k ∈ V m of the bases introduced in Proposition 3.3(i). Let m < m and assume that k + k ≤ m. Then e k+1 v k = 0 while f k+1 v k = 0 and one can write The case k + k ≥ m + 1 can be handled in the same way starting with f m−k+1 v k = 0 and e m−k+1 v k = 0.
Lemma 3.13. Let n ∈ L(T Ω ), Γ = Γ(n) be the corresponding lamination, and g Γ be given by Definition 3.9. The following identity holds: In particular, g Γ (Id) > 0 and hence g Γ does not vanish.
Proof. Recall that the SL 2 (C) F -invariant function g Γ is defined as the component of f Γ lying in the subspace V n under the isomorphism (3.10). Lemma 3.11 gives Note that the map on the right of (3.14) acts identically on V n and, under the isomorphism on top of (3.14), each of the factors V σ,e,ne V ne of V n is obtained as a symmetrization in the corresponding factor ne k=1 W σ,e,k V ⊗ne 1 of W n . The fact that this diagram commutes yields g Γ = σ∈F , e∈∂σ (n e !) −1 · π∈ σ∈F , e∈∂σ Sym (ne) Υ(π * w C(Γ,Id) ).
Remark 3.14. Since all the maps involved into (3.14) commute with the action of SL 2 (C) F , we obtain a similar commutative diagram for the invariant subspaces: Since each of the spaces V SL 2 (C) F m , m ∈ L(T Ω ), is one-dimensional, the functions g Γ form a basis in the space Fun alg (X) SL 2 (C) Fun alg (SL 2 (C) E ) SL 2 (C) F V n , which we call the Peter-Weyl basis. We are now in the position to show that the functions f Γ also form a basis in this space. . The functions f Γ form a basis in Fun alg (X) SL 2 (C) . Moreover, the change of the bases is given by a low-triangular (with respect to the partial order on the set of laminations) matrix: Proof. Since the functions g Γ form a basis in Fun alg (X) SL 2 (C) , one can write a decomposition f Γ = ∆c Γ∆ g ∆ , note thatc ΓΓ = 1 due to Definition 3.9. The diagram (3.16) is commutative, hence this decomposition must be the image of a similar decomposition of the SL 2 (C) F -invariant component of the vector w C(Γ,Id) under the isomorphism on the top of (3.16). Therefore, the coefficientc Γ∆ vanishes unless ∆ ≤ Γ. Thus the matrix (c Γ∆ ) is low-triangular and hence the inverse one is low-triangular as well. 3.5. Extension of f Γ , g Γ to (C 2×2 ) E and orthogonality on poly-balls. In this section we discuss natural extensions of functions f Γ and g Γ from SL 2 (C) E to (C 2×2 ) E . We apply the analytic Peter-Weyl theorem for the group SU 2 (C) E in order to show that thus obtained extensions G Γ,m of g Γ are orthogonal on poly-balls with respect the Euclidean measure on (C 2×2 ) E (C 4 ) E . Then, using the interpretation of V n as the spaces of homogeneous polynomials of degree n we derive an exponential lower bound for the L 2 -norms of G Γ,d on these poly-balls which is required for the further analysis performed in Section 5.
Since SL 2 (C) E is an algebraic subvariety of (C 2×2 ) E one has a trivial surjection The isomorphism (3.10) provides a way to construct a left inverse to this surjection. Namely, there is a natural extension of the action (3.9) of SL 2 (C) E on V n to an action of (C 2×2 ) E : to define the latter on a factor V ne of V n one simply extends the standard action of SL 2 (C) on V 1 to an action of C 2×2 and view V ne as the symmetrization of V ⊗ne 1 . Clearly, for each v ∈ V n , this provides a natural extension of the function Υ(v) : Note that the action (3.4) of SL 2 (C) F on SL 2 (C) E naturally extends to an action on (C 2×2 ) E and that (3.19) commutes with this action. Therefore, one has an injection Given a lamination Γ and a multi-index m = (m e ) e∈E we define

21)
Remark 3.17. By construction, G Γ,m is a homogeneous polynomial of degree |Γ| + 2|m| on (C 2×2 ) E C 4|E| , invariant under the action of SL 2 (C) F on (C 2×2 ) E . More precisely, G Γ,m is a homogeneous polynomial of degree n e + 2m e in coordinates of the space C 2×2 assigned to an edge e, for each e ∈ E. Below we call such polynomials homogeneous of multi-degree d. Proof. Note that the action SL 2 (C) F respects the degrees of homogeneity in coordinates assigned to an edge e. Therefore, it is enough to show that, given d ∈ Z E ≥0 , the set {G Γ,m | n(Γ) + 2m = d} is a basis in the space of homogeneous SL 2 (C) F -invariant polynomials of multi-degree d, see Remark 3.17.   . Since the dimensions coincide, the mapping (3.22) is an isomorphism. Using the fact that this mapping commutes with the action of SL 2 (C) F we get m : 2m≤d

Recall that each of the spaces
, is one-dimensional and the image of its generating vector is the polynomial G Γ(d−2m),m . Therefore, these polynomials indeed form a basis in the space Fun  .7)) for general matrices A e ∈ C 2×2 . This is done as follows: first, consider only non-backtracking loops γ on T • Ω ; second, replace each inverse matrix A −1 e appearing in this product by the conjugate matrix A ∨ e . The latter matrix is defined by the identity A ∨ e v∧w = v∧A e w, note that A e +A ∨ e = Tr A e ·Id.
Let hol ∨ (·, γ) be the extension of hol(·, γ) on (C 2×2 ) E defined above. It is easy to see that Tr hol ∨ (A, γ) does not depend on the orientation of γ and a choice of the starting point. This allows one to extend the functions f Γ from SL 2 (C) E to (C 2×2 ) E as note that all functions F Γ,m , m = (m e ) e∈E ∈ Z E ≥0 , coincide with f Γ on SL 2 (C) E . Lemma 3.19. For each lamination Γ and m ∈ Z E ≥0 , the following identity holds: , where the coefficients c Γ∆ are the same as in Theorem 3.15.
We now move to the analysis of functions G Γ,m as elements of the Hilbert space L 2 (B R ) on poly-balls B R = B E R , see (3.18). Recall that the Euclidean measure on B R is given by the scalar product defined on each of the components as Classically, irreducible unitary representations of SU 2 (C) are given by the restrictions of irreducible representations SL 2 (C) → End(V n ) provided that the scalar product on V n Sym V ⊗n 1 is obtained from the standard scalar product on V 1 . Let Γ 1 = Γ 2 be two distinct laminations and Since n e (Γ 1 ) = n e (Γ 2 ) for some edge e ∈ E, it follows from Theorem 3.20 that where U = (U e ) e∈E ∈ SU 2 (C) E and µ Haar (dU) = e∈E µ Haar (dU e ). The next step is to deduce the following lemma from the orthogonality condition (3.25).
Since the ball B 1 is invariant under rotations A = (A e ) e∈E → (e 2πite A e ) e∈E =: e 2πit A, in this case one easily gets Assume now that n(Γ 1 ) + 2m 1 = n(Γ 2 ) + 2m 2 , hence Γ 1 = Γ 2 as otherwise one would have (Γ 1 , m 1 ) = (Γ 2 , m 2 ). Let B + 1 := {H ∈ B 1 | H ≥ 0} be the set of non-negative Hermitian matrices H ∈ C 2×2 satisfying Tr (HH * ) < 1. Consider the mapping This mapping is a bijection modulo zero measure sets due to the existence and the uniqueness of the polar decomposition of generic matrices A ∈ B 1 . Note that the scalar product (3.24) satisfies . Therefore, the Euclidean measure on B 1 can be factorized as λ B 1 (dA(θ, U, H)) = dθ · µ Haar (dU ) · ν(dH), (3.27) for some absolutely continuous measure ν on B + 1 . Let v 1 ∈ σ∈F e∈∂σ V ne(Γ 1 ) and v 2 ∈ σ∈F e∈∂σ V ne(Γ 2 ) be such that 3.6. Exponential lower bound for the norms of functions G Γ,m . The goal of this section is to derive the following uniform lower bound for the L 2 -norms of the functions G Γ,m on poly-balls B R : There exists a (small) absolute constant η 0 > 0 such that for all laminations Γ and multi-indices m ∈ Z E ≥0 , the following estimate holds: (3.28) The constant η 0 is independent of T Ω and the number of punctures λ 1 , . . . , λ n ∈ Ω.
Remark 3.23. The proof is postponed until the end of this section. Note that G Γ,m is a homogeneous polynomial of total degree |Γ| + 2|m| and B R is a poly-ball of total dimension 8|E|. Therefore, one can assume R = 1 without loss of generality.
We need some preliminaries. Recall that the space V n can be thought of as the space C[z, w] (n) of homogeneous polynomials of degree n in two variables z, w; we fix an isomorphism V n C[z, w] (n) by identifying the monomial z k w n−k with the basis vector (x k y n−k ) sym ∈ V n , see Section 3.3. For each multi-index n ∈ Z E ≥0 , this isomorphism induces an isomorphism of the space V n and the space of homogeneous polynomials of multi-degree n in variables z = (z σ,e ) σ∈F ,e∈∂σ and w = (w e,σ ) σ∈F ,e∈∂σ , note that the total degree of these polynomials is 2|n| since V n contains two factors V σ,e,ne per edge e ∈ E.
Lemma 3.24. Given n ∈ L(T Ω ) and the lamination Γ = Γ(n), let v Γ ∈ V n be the vector corresponding to the function g Γ under the isomorphism (3.10) and P Γ ∈ C[z, w] (n) be the homogeneous polynomial corresponding to v Γ under the isomorphism (3.29). Then, where e 1 , e 2 , e 3 ∈ E denote the three edges adjacent to σ and e i+3 := e i .
Lemma 3.25. Let v 1 , v 2 ∈ V n and P 1 , P 2 ∈ C[z, w] (n) be the corresponding homogeneous polynomials of multi-degree n (and of total degree 2|n|). Then, where dλ S := σ∈F , e∈∂σ λ Sσ,e is the product of the surface measures on the spheres S σ,e .
Proof. It is enough to consider the case v 1 = v 2 =: v = σ∈F , e∈∂σ v σ,e . For such vectors the claim follows from the component-wise identity where v ∈ V n and P ∈ C[z, w] (n) is the homogeneous polynomial corresponding to v.
To prove this identity consider a mapping SU 2 (C) → S given by g → v (0) g, where v (0) = (1, 0) ∈ C 2 corresponds to the basis vector x ∈ V 1 in the notation of Section 3.3. This mapping is a diffeomorphism and the pushforward of the Haar measure on SU 2 (C) is (2π 2 ) −1 λ S . Since P corresponds to v, one has P (v (0) g) = g(x ⊗n ), v Vn . Therefore, where the last equality follows from Theorem 3.20.
Proof of Proposition 3.22. As already mentioned above, one can assume R = 1 due to homogeneity reasons. Let v Γ ∈ V n and P Γ ∈ C[z, w] (n) correspond to the function g Γ as discussed in Lemma 3.24. Using factorization (3.26) of the Euclidean measure on B 1 in the same way as in the proof of Lemma 3.21 one gets the identity Theorem 3.20 gives Vn .
Applying Lemma 3.25 we arrive at the identity where H[P Γ ] is obtained from P Γ by the change of variables (z σ,e , w σ,e ) → (z σ,e , w σ,e )H e . Finally, using the explicit form of the polynomial P Γ provided by Lemma 3.24 one can choose a (small) absolute constant ε 0 > 0 so that ≥ ε 0 and the following holds provided that Tr (H 2 e ) ≥ 1 − ε 0 for all edges e ∈ E: The desired exponential lower bound for the norm G Γ,m L 2 (B 1 ) follows easily.

Estimate of coefficients in the Fock-Goncharov theorem
The goal of this section is to derive the estimate |c ∆Γ | ≤ 4 |Γ| for the coefficients of the Fock-Goncharov change of the bases g Γ = ∆≤Γ c Γ∆ f ∆ discussed in Section 3.4. To the best of our understanding, a similar exponential bound was implicitly used in [14, p. 483] and [11, p. 957] but we were unable to find a proof of such an estimate in the existing literature. (Note that an exponential bound for the coefficients of the inverse change of the bases f Γ = ∆≤Γc Γ∆ g ∆ trivially follows from the orthogonality of g ∆ but this fact a priori does not imply an exponential estimate of c Γ∆ .) In order to study the coefficients c Γ∆ we use the well-known connection between the Kauffman skein algebra with the parameter q = −1 and the ring of invariants Fun alg (X) SL 2 (C) (e.g., see [16]): functions g Γ and f Γ correspond to some natural elements of the skein algebra and the matrix c Γ∆ admits some combinatorial description through this correspondence. One of the key ingredients of the proof given below is a recent result due to D. Thurston [28] on the positivity of the so-called bracelet basis in the skein algebra with q = 1, see Section 4.4 for more details. Another input is a representation of the skein algebra in the space of Laurent polynomials via Thurston's shear coordinates on the moduli space of hyperbolic structures on Ω\{λ 1 , . . . , λ n }, see Section 4.3. These ideas were kindly communicated to the authors by Vladimir Fock and we believe that a core part of the proof of Theorem 4.9 should in fact be credited to him.
Several parts of the material presented in this section are classical or well-known. As in Section 3 we collect all them together in order to introduce a consistent notation and for the sake of completeness. ). We say that an (oriented) knot is trivially framed if v 1 is the tangent vector to the knot and v 3 is everywhere vertical, each framed knot is equivalent to a trivially framed one up to a framed isotopy. Denote by L unknot the trivially framed unknot associated with a simple contractible loop in Σ.
Let W be a C-vector space generated by framed isotopy classes of framed links. We define a multiplication of two links L 1 and L 2 by placing L 1 under L 2 (with respect to the direction of the projection [0, 1] × Ω → [0, 1]) and taking the union. This makes W into an algebra with the unit represented by the empty link. We say that three framed links L 1 , L 0 and L ∞ form a Kauffman triple if they can be drawn identically except in a ball where they appear as shown in Figure 4. Let I be the two-sided ideal in W generated by the relations L 1 − qL 0 − q −1 L ∞ = 0 for all Kauffman triples and L unknot +q 2 +q −2 = 0. The Kauffman skein algebra with the parameter q is the quotient Sk(M, q) := W/I.
In our paper we are interested in the particular cases q = ±1 only. Note that the relations in I immediately imply that the algebras Sk(M, ±1) are commutative.The following theorem is due to Przytyscki and Sikora [19], see also [5] for a close result.  These observations can be completed to a proof of the fact that the mapping ψ is a well-defined homomorphism of algebras as it is done in [5, proof of Theorem 3].
One can now use the Fock-Goncharov theorem to prove that ψ is an isomorphism. Namely, each lamination Γ naturally gives rise to a framed link in M : represent Γ as a disjoint union of simple curves and attach a trivial frame to each of these curves. It is easy to see that the image of thus obtained framed link under ψ equals (up to a sign) to f Γ and that such links span Sk(M, −1). Using the fact that f Γ is a basis of Fun alg (X) SL 2 (C) one concludes that ψ is an isomophism.

4.2.
Skein algebra with q = 1 and twisted representations. It is well known that the skein algebras Sk(M, −1) and Sk(M, 1) are closely related via spin structures on Σ, below we briefly discuss this correspondence.
Recall that the unit tangent bundle U Σ of Σ is defined as where 0 stands for the zero section of T Σ. Since π 2 (Σ) = 0, one has a short exact sequence π 1 (S 1 ) → π 1 (U Σ) → π 1 (Σ) and concludes that π 1 (U Σ) is a central extension of π 1 (Σ) by Z. Let z ∈ π 1 (U Σ) denote the image of the generator of Z. A spin structure on Σ is an element ξ ∈ H 1 (U Σ, Z 2 ) such that ξ(z) = 1. For a trivially framed knot K, let γ K be a loop in U Σ given by points of γ K ⊂ Σ and projections of the first (tangential) vector of the framing. It is easy to see that isotopic framed knots define homotopic curves in U Σ.
It is also well known that each spin structure ξ on a compact orientable surface Σ can be given by a vector field V ξ on Σ with isolated zeroes of even index: if K is a trivially framed knot, then where wind(γ K ; V ξ ) denotes the total rotation angle of the tangent vector to γ K measured with respect to V ξ . In fact, below one can always assume that V ξ is constant (which corresponds to the unique spin structure on the original simply connected domain Ω ⊃ Σ), so that wind(γ K ; V ξ ) = wind(γ K ) is the usual total rotation angle of the tangent vector along the loop γ K in Ω.
where the product is taken over all components K of a framed link L, is a correctly defined isomorphism of algebras.
We now introduce an additional terminology that will be used in Section 4.3.
Proof. This is a trivial corollary of Theorem 4.1, Lemma 4.2 and definition (4.1).
Remark 4.5. It follows from Proposition 4.4 that each twisted representation ρ ∈ X twist gives rise to an algebra homomorphism (evaluation) Ψ ρ : Sk(M, 1) → C defined on trivially framed knots as K → Tr ( ρ( γ K )) and extended by additivity and multiplicativity.

4.3.
Twisted representations associated with hyperbolic structures on Σ and the representation of Sk(M, 1) in the algebra of Laurent polynomials. In this section we discuss the class of twisted representations ρ x coming from hyperbolic structures on Σ parameterized by Thurston's shear coordinates x = (x e ) e∈E ∈ R E + . In particular, we show that all functions x → Tr ( ρ x ( γ K )) associated with trivially framed knots K ∈ Sk(M, 1) are Laurent polynomials in the variables √ x e with sign coherent (simultaneously positive or simultaneously negative) coefficients, the last observation was pointed out to us by Vladimir Fock. We discuss only the case when Σ = Ω \ {λ 1 , . . . , λ n } is a punctured simply connected domain, see [3] for the general construction in the case of surfaces with free non-abelian fundamental group. Let π : Σ → Σ be the universal cover of Σ. Classically, given a hyperbolic metric on Σ one can consider a uniformization f : H → Σ where H := {z ∈ C : Imz > 0} stands for the upper complex plane, note that f is defined uniquely up to Möbius automorphisms of H. Provided that the edges of the triangulation T Ω of Σ are drawn as hyperbolic geodesics and T Ω := π −1 (T Ω ) denotes the corresponding triangulation of Σ, the following properties hold: (i) the preimage under f of each triangle of T Ω is an ideal triangle in H; (ii) deck transformations of Σ form a discrete subgroup of the group Aut(H). Recall that Aut(H) ∼ = PSL 2 (R) (automorphisms of H are given by Möbius transforms z → (az + b)/(cz + d) with a, b, c, d ∈ R defined up to a common multiple) and that for each (z 0 , v 0 ), (z 1 , v 1 ) ∈ U H there exists a unique automorphism g ∈ Aut(H) such that g(z 0 ) = z 1 and the vector g (z 0 )v 0 has the same direction as v 1 .
Given a uniformization f : H → Σ and a base point (z 0 , v 0 ) ∈ U H, one can construct a twisted representation ρ f as follows. For a smooth loop γ : [0, 1] → U Σ such that γ(0) = γ(1) = (π • f )(z 0 , v 0 ), let γ : [0, 1] → U Σ denote its lift on U Σ and let g γ (t) ∈ Aut(H) sends (f −1 • γ)(0) = (z 0 , v 0 ) to (f −1 • γ)(t). One can now lift the path g γ : [0, 1] → Aut(H) ∼ = PSL 2 (R) to g γ : [0, 1] → SL 2 (R) so that g γ (0) = Id and set note that this mapping is nothing but a lift to SL 2 (R) of the automorphism of H corresponding to the deck transformation of Σ given by the projection of γ onto Σ. In particular, ρ f does not depend on the choice of the lift γ of γ. It is straightforward to check that ρ f is a twisted representation on Σ. Moreover, the change of the base point (z 0 , v 0 ) and/or of the uniformization f amounts to a SL 2 (C)-conjugation of ρ f . We now introduce Thurston's shear coordinates on the space of hyperbolic structures on Σ. Given a collection x = (x e ) e∈E of positive parameters associated to the edges of the triangulation T Ω , one can construct a hyperbolic structure on Σ such that the following property holds for (any of) the corresponding uniformizations f x : H → Σ: (iii) for each edge e of T Ω , the quadrilateral σ left (e) ∪ e ∪ σ right (e) formed by two faces adjacent to e is conformally equivalent to the ideal quadrilateral R(x e ) ⊂ H with vertices −1, 0, x e , ∞ so that the edge e corresponds to the line √ −1 · R + ⊂ R(x e ). In fact, one can use property (iii) to draw a preimage of the triangulation T Ω in H. To do this, start with two faces adjacent to an arbitrarily chosen edge and draw them so that they form the ideal rectangle R(x e ) and then extend this drawing step by step so that each face of T Ω is drawn as an ideal triangle and the property (iii) is satisfied. As a result of this procedure one obtains a diffeomorphism f x : H → Σ which can be used to define the desired hyperbolic structure on Σ and to project it onto Σ. It can be easily shown that each hyperbolic structure on Σ can be obtained in this manner but we do not need this fact in our paper.
Below we denote by ρ x the twisted SL 2 (C) representation corresponding to the hyperbolic metric on Σ and the uniformization f x : H → Σ constructed above. Recall that ρ x is defined up to a SL 2 (C)-conjugation only as we do not fix neither a base point (x 0 , v 0 ) ∈ H nor the starting edge of the construction. Nevertheless, following Proposition 4.4 and Remark 4.5, for each x ∈ R E + one can correctly define the evaluation which acts on trivially framed knots as Ψ x (K) = Tr ( ρ x ( γ K )) and is extended to the skein algebra Sk(M, 1) by linearity and multiplicativity.
The following proposition is the key result of this section. Recall that, given a minimal smooth multi-curve C on Σ = Ω \ {λ 1 , . . . , λ n } (see section 3.4 for the definition) one can obtain an element L(C) ∈ Sk(M, 1) by framing all the components of C trivially and then resolving the intersections in an arbitrary way: the resulting link depends on the choice of these resolutions but its class in Sk(M, 1) does not. Proof. Since for each framed link L there exists a trivially framed link L obtained from a minimal multi-curve such that L = ±2 m L in Sk(M, 1) for some m ≥ 0, one can assume that L is obtained from a minimal multi-curve. Moreover, since Ψ x is an algebra homomorphism one can further assume that L = L(γ) is a trivially framed knot obtained from a minimal (i.e. having no nugatory self-crossings, see Fig. 3) loop γ.
One can now replace γ by the concatenation of smooth segments γ(e i , e i+1 ) crossing the edges of E orthogonally and running from a i ∈ e i to a i+1 ∈ e i+1 inside the face σ i+1 . For each i = 0, . . . , m − 1, (a) either the preimage ϕ −1 i (e i+1 ) of the next edge crossed by γ connects x e i and ∞, in this case we say that γ makes a counterclockwise turn inside σ i+1 , see Fig. 5,(b) or this preimage of e i+1 connects 0 and x e i : a clockwise turn inside σ i+1 .
clockwise turn Figure 5. The piece γ(e i , e i+1 ) of a loop γ in the chart ϕ −1 i : Σ → C.
One easily sees that the composition mapping ϕ −1 i •ϕ i+1 (which maps the line √ −1·R + onto the preimage ϕ −1 i (e i+1 )) is given by z → x e i · (z + 1) in the case (a), z → x e i z z + 1 in the case (b), moreover the lift of the corresponding path in Aut(H) ∼ = PSL 2 (R) to SL 2 (R) leads to the following matrix A i representing the Möbius automorphism ϕ −1 i • ϕ i+1 : in the case (b).
Let γ be the loop in the unit tangent bundle U Σ corresponding to γ and γ(e i , e i+1 ) be its segment corresponding to γ(e i , e i+1 ). From now onwards we fix the uniformization f := ϕ 0 : H → Σ. According to the definition (4.2) we have is a Laurent polynomial in the variables ( √ x e ) e∈E with positive integer coefficients. In particular we have proved (ii). The argument given above defines a required Laurent polynomial P L for each element L = L(γ) ∈ Sk(M, 1) obtained from a minimal loop γ. This definition can be then extended to the whole skein algebra Sk(M, 1) by linearity and multiplicativity. Finally, since the mapping L → P L ((t e ) e∈E ) is an algebra homomorphism for each specification t e := √ x e , the same is also true for formal variables t e .  Proof. See [28,Theorem 2].
Recall that we denote by |C| the minimal number of intersections of a multi-curve homotopic to C with the edges of the triangulation T Ω and by Γ(C) the corresponding lamination, see Section 3.4 for more details. The next lemma is a simple combination of the positivity results provided by Proposition 4.6(ii) and Theorem 4.7.
Proof. As each multi-curve is homotopic to a minimal one up to nugatory self-crossings and homotopically trivial components one can assume that C is minimal. Let P L(C) be the Laurent polynomial given by Proposition 4.6 and x = 1 ∈ R E + be the vector consisting of units: x e = 1 for all e ∈ E. It follows from the construction of P L(C) (see (4.4)) that P L(C) (1) ≤ 2 |C| . Applying the evaluation Ψ 1 to the identity (4.6) and noting that P L(b(Γ)) (1) ≥ 1 one obtains the desired estimate.
We are now in the position to prove the main result of this Section. Recall that we are interested in an exponential upper bound for the coefficients of the Fock-Goncharov change of the bases Proof. Recall that the function f ∆ ∈ Fun alg (X) SL 2 (C) corresponds to ±L(∆) ∈ Sk(M, 1) under the isomorphisms provided by Theorem 4.1 and Lemma 4.2. Due to Lemma 3.13, the function g Γ correspond to the averaging of the signed knots ±L(C(Γ, π)) over permutations π ∈ σ∈F , e∈∂σ Sym (n e ). Let Then, c Γ∆ is equal to the average of the signed coefficients β C∆ over the permutations π and it is enough to prove that |β C∆ | ≤ 4 |C| = 4 |Γ| for each multi-curve C = C(Γ, π).
Using the bracelet basis of Sk(M, 1) discussed above one writes Now one can expand each of the bracelet links L(b(Γ )) ∈ Sk(M, 1) via the functions L(∆) with ∆ ⊂ Γ using (4.5): where each coefficient α Γ ∆ is the product over homotopically distinct loops of Γ of some coefficients of the Chebyshev polynomials (corresponding to these loops) Using the trivial upper bound 2 m for these coefficients one obtains the crude estimate |α Γ ∆ | ≤ 2 |Γ | ≤ 2 |Γ| . Therefore, Remark 4.10. It is worth noting that the constant 4 in the exponential upper bound provided by Theorem 4.9 is far from being optimal. For instance one can easily improve the constant 2 in Lemma 4.8 to 1 2 ( √ 5 + 1), which is equal to the L 2 -norm of the matrices A i with x e = 1, not speaking about a huge overkill in the estimate of coefficients α Γ ∆ used above.

Proof of the main results
This section is organized as follows. In Section 5.1 we discuss the expansions of SU 2 (C) F -invariant functions on poly-balls B R ⊂ SL 2 (C) E via the functions F Γ,m constructed in Section 3 (see (3.21)). Note that the exponential estimates provided by Proposition 3.22 and Theorem 4.9 play a crucial role in the proof of Lemma 5.3. We then pass to the analysis of holomorphic functions on the variety Z unip := { A ∈ SL 2 (C) E | Tr (hol(A, (λ i ))) = 2 for all i = 1, . . . , n } (5.1) (here and below, the holomorphicity condition is always required on the regular part of the variety only). Section 5.2 is devoted to extensions of holomorphic functions from Z unip ∩ B R to B R and is based upon Manivel's Ohsawa-Takegoshi type theorem, it is needed for the existence part of Theorem 5.11. Section 5.3 contains a version of Hilbert's Nullstellensatz for holomorphic functions vanishing on Z unip ∩ B R , we need this for the uniqueness part of Theorem 5.11. Finally, Section 5.4 is devoted to the proof of our main result, Theorem 5.11, on the existence and uniqueness of expansions of holomorphic functions on X unip in the basis f Γ indexed by macroscopic laminations Γ (i.e., those not containing any of the loops (λ i ) surrounding only one puncture).

Expansions of SU
Recall that the action of the group SL 2 (C) F on the space (C 2×2 ) E is given by (3.4).
We start with a simple preliminary lemma.
Lemma 5.1. Let a holomorphic function F : (C 2×2 ) E → C be invariant under the action of SU 2 (C) F ⊂ SL 2 (C) F . Then, F is also invariant under the action of SL 2 (C) F .
(ii) Grouping the terms in (5.2) according to their degree of homogeneity d = |Γ|+2|m|, one sees that each term in the series The next lemma shows that each SU 2 (C) F -invariant holomorphic function defined on a poly-ball B R necessarily admits an expansion (5.2) in a smaller poly-ball B1 5 η 0 R . Recall that the absolute constant η 0 is introduced in Proposition 3.22. Lemma 5.3. Given r > 0, denote R := 5η −1 0 r. Let F ∈ L 2 (B R ) be an SU 2 (C) Finvariant holomorphic function on B R . Then, there exist coefficients p Γ,m indexed by laminations Γ and tuples m ∈ Z E ≥0 such that the following holds: for all Γ, m, and Proof. Let F (A) = d≥0 F (d) (A) be the Taylor expansion of the function F at the origin. Since this expansion is unique, the homogeneous polynomials F (d) of degree d are SU 2 (C) F -invariant. Lemma 5.1 implies that these polynomials are also SL 2 (C) Finvariant so one can expand them in the basis G ∆,k with |∆| + 2|k| = d due to Lemma 3.18. More precisely, Lemma 3.21 yields . Due to Proposition 3.22 we have the following estimate: . We now use the expansion G ∆,k = Γ≤∆ c ∆Γ F Γ,k+ 1 2 (n(∆)−n(Γ)) provided by Lemma 3.19, which leads to the following identity: One can now easily deduce (5.3) from the estimate |c ∆Γ | ≤ 4 |∆| provided by Theorem 4.9, the estimate of coefficients q ∆,k given above and the following crude bound: The exponential upper bound (5.3) and Lemma 5.2(i) immediately imply (5.4).

5.2.
Holomorphic extensions from Z unip ∩ B R to B R . This section relies upon the following particular case of Manivel's Ohsawa-Takegoshi type theorem [15]; see also a very helpful introduction to this subject due to Demailly [6]. Then, each holomorphic function f : Z reg := Z \ Z sing → C such that Z |f | 2 |Λ r ds| −2 ω k−r < ∞, admits a holomorphic extension F : where ω = i 2 e∈E Tr (dA e ∧ dA * e ) and the constant C r,k > 0 depends only on r and k.
which holds true for all one-dimensional restrictions z → s(A + zB) of s.
We now apply Theorem 5.5 to the variety Z unip ⊂ (C 2×2 ) E defined by (5.1). For our purposes it is enough to assume that the function f is bounded, we also do not need the sharp weight (−|s| r log |s|) −2 ≥ 1 in the L 2 -norm of its extension F . Proposition 5.6. Let R > √ 2 and f : Z reg unip ∩B R → C be a bounded holomorphic function. Then, there exists a holomorphic function F : , for some constant depending on R and the triangulation T Ω but independent of f .
Proof. Denote s := ((F ∅,n e − 1) e∈E , (F (λ i ) − 2) n i=1 ) : (C 2×2 ) E → C |E|+n and fix a small positive constant c(R, T Ω ) so that c(R, T Ω ) · |s| ≤ e −1 on B R . Due to Theorem 5.5, the function f admits a holomorphic extension F from Z reg unip ∩ B R to B R such that Therefore, it is enough to check that |Λ r ds| −2 ∈ L 1 loc (Z unip ). Let T ⊂ E, |T | = n, be a spanning tree (on n + 1 vertices ∂Ω, λ 1 , . . . , λ n ) of the triangulation T Ω such that the graph T \ {λ k , . . . , λ n } is connected for all k = n, . . . , 1. Note that the mapping (More accurately, one should multiply det B i by several factors (det A e ) −1 corresponding to the edges incident to λ i and possibly by factors (det B j ) −1 , j > i, coming from earlier steps of the reconstruction of (A e ) e∈T , but all these additional factors do not affect Λ r ds on SL 2 (C) E ). As the gradient of A → det A does not vanish on SL 2 (C) there is nothing to check for the first coordinates (A e ) e∈E\T . For the coordinates (B i ) n i=1 , the only degeneracy of ds 0 ∧ ds 0 on s −1 0 (0) is at B = Id. Writing B = t − ix y + iz −y + iz t + ix , t, x, y, z ∈ C, ω = i 4 (dt ∧ dt + dx ∧ dx + dy ∧ dy + dz ∧ dz), it remains to check that t=1, x 2 +y 2 +z 2 =0, |x|,|y|,|z|≤1 ω ∧ ω |x| 2 + |y| 2 + |z| 2 < ∞, which is straightforward.
Remark 5.7. It is easy to see that Proposition 5.6 remains true (with exactly the same proof) if one replaces the variety Z unip by Z k := {A ∈ SL 2 (C) E | Tr (hol(A, (λ i ))) = 2 for all i = 1, . . . , k} (5.6) (note that Z unip = Z n ⊂ Z n−1 ⊂ · · · ⊂ Z 1 ⊂ Z 0 = SL 2 (C) E ). Remark 5.9. (i) It is worth noting that, if F were a polynomial, then the result would follow from the classical Hilbert's Nullstellensatz as one can easily check that the radical of the ideal I = (det A e − 1) e∈E , (Tr hol ∨ (A, (λ i )) − 2) n i=1 coincides with I. (ii) Although one can prove Proposition 5.8 by means of algebraic geometry, below we prefer to take advantage of the analytic tools already introduced in the previous section (namely, Proposition 5.6 and Remark 5.7) and the following simple lemma. Proof. If Id / ∈ V , this fact easily follows from the local analysis since Y ∩ V is smooth in V and dt does not vanish there. In the opposite case, let U V be a small neighborhood of Id ∈ V . The same local analysis implies that the ratio f /t can be viewed as a holomorphic function on V \ U and the Hartogs extension theorem guarantees the existence of its holomorphic extension in the whole V .
Proof of Proposition 5.8. Let Z unip = Z n ⊂ Z n−1 ⊂ . . . ⊂ Z 1 ⊂ Z 0 = SL 2 (C) E be given by (5.6) and B i := hol(A, (λ i )) as in the proof of Proposition 5.6; recall that each Z k is cut off by the equations t 1 = . . . = t k = 0, where t i := Tr B i − 2.
Consider the meromorphic function F/t n on Z n−1 ∩ B R and recall the coordinates (5.5) from the proof of Proposition 5.6. If one fixes all (A e ) e∈E\T and (B i ) n−1 i=1 , then Lemma 5.10 guarantees that the ratio F/t n is a holomorphic function of the last coordinate B n . Therefore, there exists a holomorphic function h n : Z n−1 ∩ B R → C such that F = t n · h n on Z n−1 ∩ B R . Due to Remark 5.7, for each R n−1 < R one can find a holomorphic extension H n : B R n−1 → C of the (bounded on Z n−1 ∩ B R n−1 ) function h n . As t(B n ) = Tr hol(A, (λ n )) − 2 on SL 2 (C) E , one has and one can iterate this procedure considering the ratio F n−1 /t n−1 on Z n−2 ∩ B R n−2 with R n−2 < R n−1 < R n , etc. After n steps, one obtains the existence of holomorphic functions H i : where R 0 can be chosen so that R < R 0 < R.
In order to complete the proof and to construct the required functions (H e ) e∈E in addition to (H i ) n i=1 one simply repeats the same arguments for the collection of functions (det A e − 1) e∈E instead of (t i ) n i=1 . Since the manifold SL 2 (C) E ⊂ (C 2×2 ) E is smooth, there is even no need in an analogue of Lemma 5.10 along this procedure.
Proof. To prove the existence of coefficients p Γ consider a holomorphic function F : B R → C obtained by applying Proposition 5.6 to the function f • φ : Z unip ∩ B R → C.
Recall that one has Note that we can in addition assume that the function F is invariant under the action of SU 2 (C) F on B R ⊂ SL 2 (C) E . Indeed, one can always replace F by its average F SU 2 (C) F over the orbits of this action: the norm of F in L 2 (B R ) does not increase under this averaging due to (3.27) and one still has F SU 2 (C) F = f • φ on Z unip ∩ B R since the mapping φ is invariant under the action of SL 2 (C) F . Thus we can use Lemma 5.3 to expand F in B r as where |p Γ,m | ≤ r −(|Γ|+2|m|+4|E|) · F L 2 (B R ) and the series converges absolutely and uniformly on each smaller poly-ball B r , r < r.
Note that each lamination can be represented as a disjoint union of a macroscopic lamination Γ, k 1 copies of (λ 1 ), k 2 copies of (λ 2 ) and so on. For shortness, below we use the notation Γ (λ k ) to describe such a lamination, where k = (k i ) n i=1 ∈ Z n ≥0 . By definition, Since one has det A e = 1 and F (λ i ) (A) = 2 for A ∈ Z unip , we get the identity where p Γ := k∈Z n ≥0 ,m∈Z E ≥0 p Γ (λ k ),m · 2 |k| . As n(Γ (λ k )) ≥ n(Γ) + 2|k|, it is easy to see that This gives the desired exponential upper bound (5.7) for |p Γ |. We now move on to the uniqueness of expansion (5.8). Assume that two sequences of coefficients p Γ ,p Γ satisfy the upper bound |p Γ |, |p Γ | = O(r −|Γ| ) as |Γ| → ∞ and that the corresponding series (5.8) coincide on D r . Denote Recall that this series converges absolutely and uniformly on compact subsets of B r due to Lemma 5.2(i). Therefore, F is a holomorphic function on B r which vanishes along Z unip . Due to Proposition 5.8 one can find bounded holomorphic functions (H e ) e∈E and (H λ i ) 1≤i≤n , defined on B r , r < r, such that Moreover, since all the functions F , F ∅,n e and F (λ i ) are SU 2 (C) F -invariant, one can assume that all the functions H α , α ∈ E ∪{λ 1 , . . . , λ n } are invariant as well by averaging each of them over orbits of the action of the compact group SU 2 (C) F on A ∈ B r . We proceed with expanding these functions in F Γ,m as provided by Lemma 5.3. Namely, for each < 1 5 η 0 r and α ∈ E ∪ {λ 1 , . . . , λ n } one has H α (A) =