On a Nielsen–Thurston classification theory for cluster modular groups

Ishibashi, Tsukasa

doi:10.5802/aif.3250

Cited by 7 publications

(9 citation statements)

References 30 publications

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“…Next let us turn our attention to the problem finding a good generators of cluster modular groups. As a candidate for an appropriate class of generators, the cluster Dehn twists has been introduced in the author's previous work [15]. He proved that the cluster Dehn twists have a similar dynamical behavior to that of Dehn twists in mapping class groups.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists

Ishibashi

2020

SIGMA

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists

Ishibashi

2020

SIGMA

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“…Based on this correspondence, the first author gave an analogue of the Nielsen-Thurston classification for a general cluster modular group in [23], which classifies the mutation loops into three types: periodic, cluster-reducible and cluster-pseudo-Anosov. However there exists a slight discrepancy between pseudo-Anosov and cluster-pseudo-Anosov even for a mutation loop given by a mapping class: a pseudo-Anosov mapping class provides a cluster-pseudo-Anosov mutation loop, but the converse is not true.…”

Section: Sign Stability and The Main Theoremmentioning

confidence: 99%

Algebraic entropy of sign-stable mutation loops

Ishibashi

Kano

2021

Geom Dedicata

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In the theory of cluster algebras, a mutation loop induces discrete dynamical systems via its actions on the cluster Aand X -varieties. In this paper, we introduce a property of mutation loops, called the sign stability, with a focus on the asymptotic behavior of the iteration of the tropical X -transformation. The sign stability can be thought of as a cluster algebraic analogue of the pseudo-Anosov property of a mapping class on a surface. A sign-stable mutation loop has a numerical invariant which we call the cluster stretch factor, in analogy with the stretch factor of a pseudo-Anosov mapping class on a marked surface. We compute the algebraic entropies of the cluster Aand X -transformations induced by a sign-stable mutation loop, and conclude that these two coincide with the logarithm of the cluster stretch factor. This gives a cluster algebraic analogue of the classical theorem which relates the topological entropy of a pseudo-Anosov mapping class with its stretch factor.

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“…If χ(T n,w ) = 0 then the cluster modular group of a type T n,w cluster algebra is exactly Γ Tn,w . Let (12) i odd = {i j |3 ≤ j ≤ n i , j odd} and i even = {i j |3 ≤ j ≤ n i , j even}.…”

Section: 1mentioning

confidence: 99%

Cluster Modular Groups of Affine and Doubly Extended Cluster Algebras

Kaufman,

Greenberg

2021

Preprint

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We calculate the cluster modular groups of affine and doubly extended type cluster algebras in a uniform way by introducing a new family of quivers. We use this uniform description to construct a natural finite quotient of the cluster complex of each affine and doubly extended cluster algebra. Using this construction, we introduce the notion of affine and doubly extended generalized associahedra, and count their facets. Contents 1. Introduction 1 2. Preliminaries on Dynkin Diagrams, Quivers, and Mutations 3 3. The Cluster Modular Group 8 4. Type T n,w Cluster Algebras 11 5. Affine Cluster Algebras 15 6.

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On a Nielsen–Thurston classification theory for cluster modular groups

Abstract: 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.- .

Cited by 7 publications

References 30 publications

Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists

Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists

Algebraic entropy of sign-stable mutation loops

Cluster Modular Groups of Affine and Doubly Extended Cluster Algebras

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