For a marked surface Σ without punctures, we introduce a skein algebra S q sl3,Σ consisting of sl 3 -webs on Σ with the boundary skein relations at marked points. We realize a subalgebra CA q sl3,Σ of the quantum cluster algebra quantizing the function ring O cl (A SL3,Σ ) inside the skein algebra S q sl3,Σ . We also show that the skein algebra S q sl3,Σ is contained in the corresponding quantum upper cluster algebra by giving a way to obtain cluster expansions of sl 3 -webs. Moreover, we show that the bracelets and the bangles along an oriented simple loop in Σ give rise to quantum GS-universally positive Laurent polynomials.Contents 42 Appendix A. Relation to the cluster varieties 50 References 52

For a symmetrizable Kac-Moody Lie algebra g, we construct a family of weighted quivers Q m (g) (m ≥ 2) whose cluster modular group Γ Qm(g) contains the Weyl group W (g) as a subgroup. We compute explicit formulae for the corresponding cluster Aand X -transformations. As a result, we obtain green sequences and the cluster Donaldson-Thomas transformation for Q m (g) in a systematic way when g is of finite type. Moreover if g is of classical finite type with the Coxeter number h, the quiver Q kh (g) (k ≥ 1) is mutation-equivalent to a quiver encoding the cluster structure of the higher Teichmüller space of a once-punctured disk with 2k marked points on the boundary, up to frozen vertices. This correspondence induces the action of direct products of Weyl groups on the higher Teichmüller space of a general marked surface. We finally prove that this action coincides with the one constructed in [GS16] from the geometrical viewpoint. Contents 1. Introduction 1 2. Notation and definitions in cluster algebra 5 3. Weyl group action 11 4. Quivers corresponding to reduced words 24 5. Application to the higher Teichmüller theory 30 6. Relation with the D g -quiver 52 Appendix A. Description of functions on Conf 3 A G 62 References 66

Les Annales de l'institut Fourier sont membres du Centre Mersenne pour l'édition scienti que ouverte www.centre-mersenne.org Tsukasa I On a Nielsen-Thurston classi cation theory for cluster modular groups Tome , n o (), p.- .

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For a marked surface Σ and a semisimple algebraic group G of adjoint type, we study the Wilson line function g [c] : P G,Σ → G associated with the homotopy class of an arc c connecting boundary intervals of Σ. We show that g [c] defines a morphism of algebraic stacks with respect to the algebraic structure on P G,Σ investigated by Shen [She20]. Combining with Shen's result, we show that the cluster Poisson algebra with respect to the natural cluster Poisson structure on P G,Σ coincides with the ring of global functions with respect to the Betti structure. Moreover we show that the matrix coefficients c) give Laurent polynomials with positive integral coefficients in the Goncharov-Shen coordinate system associated with any decorated triangulation of Σ, for suitable f and v. Pt G,Σ after Claudius Ptolemy, is isomorphic to the cluster Poisson algebra [She20, Theorem 1.1]. Here is our first result: Theorem 1 (Theorem 3.18). The Wilson line along an arc class [c] : E in → E out defines a morphism g [c] : P Pt G,Σ → G of Artin stacks. Given a free loop |γ| on Σ, by cutting the surface along an edge that intersects |γ| we get another marked surface and an arc class [c] obtained from |γ|. Based on this obsevation, one can deduce: Theorem 2. The Wilson loop along a free loop |γ| ∈ π(Σ) defines a morphism ρ |γ| : P Pt G,Σ → [G/AdG] of Artin stacks. Here [G/AdG] denotes the quotient stack of G with respect to the conjugation action on itself. As a direct consequence of Theorem 1, we can compare the two stack structures on P G,Σ : Theorem 3 (Theorem 3.25). We have an open embedding P Pt