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Fixed point theorem for cluster modular groups

Tsukasa Ishibashi

Abstract

We prove that any finite subgroup $G \subset Γ_{\boldsymbol{s}}$ of the cluster modular group has fixed points in the cluster manifolds $\mathcal{A}_{\boldsymbol{s}}(\mathbb{R}_{>0})$ and $\mathcal{X}_{\boldsymbol{s}}(\mathbb{R}_{>0})$ under a certain condition. This generalizes Kerckhoff's Nielsen realization theorem [Ker83] for the mapping class group action on the Teichmüller space. The condition holds whenever $Γ_{\boldsymbol{s}}$ admits a cluster DT transformation, and it can be also verified for all finite mutation types except for $X_7$. Our proof closely follows Kerckhoff's argument, based on the convexity of log-cluster variables.

Fixed point theorem for cluster modular groups

Abstract

We prove that any finite subgroup of the cluster modular group has fixed points in the cluster manifolds and under a certain condition. This generalizes Kerckhoff's Nielsen realization theorem [Ker83] for the mapping class group action on the Teichmüller space. The condition holds whenever admits a cluster DT transformation, and it can be also verified for all finite mutation types except for . Our proof closely follows Kerckhoff's argument, based on the convexity of log-cluster variables.
Paper Structure (18 sections, 29 theorems, 66 equations, 2 figures, 2 tables)

This paper contains 18 sections, 29 theorems, 66 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Nielsen realization theorem
  3. Nielsen realization theorem for cluster realizations of Weyl groups
  4. Convex functions
  5. Midpoint convexity
  6. Convexity of Log-Laurent functions
  7. Minima
  8. Cluster algebra
  9. Seeds and mutations
  10. Cluster manifolds
  11. Cluster modular group
  12. Cluster ensemble
  13. Nielsen realization theorem
  14. Nielsen realization theorem
  15. Step 1: Restriction to the fiber direction.
  16. ...and 3 more sections

Key Result

Theorem 1

Assume that there exists a filling set $\Lambda \subset \mathscr{U}^+_{\boldsymbol{s}}$ (def:filling_set). Then, any finite subgroup $G \subset \Gamma_{\boldsymbol{s}}$ has a fixed point in $\mathcal{A}_{\boldsymbol{s}}(\mathbb{R}_{>0})$. In particular, it also has a fixed point in $p(\mathcal{A}_{\

Figures (2)

  • Figure 1: The function $L_G$ for type $A_2$.
  • Figure 2: A quiver in the mutation class of type $X_7$.

Theorems & Definitions (57)

  • Theorem 1: \ref{['thm:fixed_point_filling']}
  • Theorem 2: \ref{['cor:fixed_point_DT']}
  • Theorem 3: \ref{['thm:fixed_point_finite_mutation']}
  • Theorem 4: \ref{['thm:Weyl_fixed_point']}
  • Lemma 1
  • proof
  • Corollary 1
  • proof
  • Lemma 2: rational convexity
  • proof
  • ...and 47 more