Presentations of cluster modular groups and generation by cluster dehn twists

Tsukasa Ishibashi

doi:10.3842/SIGMA.2020.025

Presentations of cluster modular groups and generation by cluster dehn twists

Research output: Contribution to journal › Article › peer-review

Abstract

We give a method to compute presentations of saturated cluster modular groups. Using this, we obtain finite presentations of the saturated cluster modular groups of finite mutation type X₆ and X₇. We verify that the cluster modular groups of finite mutation type Ẽ₆, Ẽ₇, Ẽ₈, G⁽ ^∗ ^, ^∗ ⁾ ₂, X₆ and X₇ are virtually generated by cluster Dehn twists.

Original language	English
Article number	025
Journal	Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
Volume	16
DOIs	https://doi.org/10.3842/SIGMA.2020.025
Publication status	Published - 2020
Externally published	Yes

Keywords

Cluster algebras
Cluster modular groups
Mapping class groups
Quivers of finite mutation type

ASJC Scopus subject areas

Analysis
Mathematical Physics
Geometry and Topology

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10.3842/SIGMA.2020.025

Cite this

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title = "Presentations of cluster modular groups and generation by cluster dehn twists",

abstract = "We give a method to compute presentations of saturated cluster modular groups. Using this, we obtain finite presentations of the saturated cluster modular groups of finite mutation type X6 and X7. We verify that the cluster modular groups of finite mutation type Ẽ6, Ẽ7, Ẽ8, G( ∗ , ∗ ) 2, X6 and X7 are virtually generated by cluster Dehn twists.",

keywords = "Cluster algebras, Cluster modular groups, Mapping class groups, Quivers of finite mutation type",

author = "Tsukasa Ishibashi",

note = "Funding Information: The author would like to express his gratitude to his supervisor, Nariya Kawazumi for continuous encouragement during this work. He is very grateful to Travis Scrimshaw for pointing out an error in the presentation of {\^Γ}X6 in the first version of this paper. Most of computations in this paper are done by using the Java applet for quiver mutations provided by Bernhard Keller, which is available at http://www.math.lsa.umich.edu/~fomin/cluster.html. This work is partially supported by JSPS KAKENHI Grant Number 18J13304 and the program for Leading Graduate School, MEXT, Japan. Publisher Copyright: {\textcopyright} 2020, Institute of Mathematics. All rights reserved.",

year = "2020",

doi = "10.3842/SIGMA.2020.025",

language = "English",

volume = "16",

journal = "Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)",

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AU - Ishibashi, Tsukasa

N1 - Funding Information: The author would like to express his gratitude to his supervisor, Nariya Kawazumi for continuous encouragement during this work. He is very grateful to Travis Scrimshaw for pointing out an error in the presentation of Γ̂X6 in the first version of this paper. Most of computations in this paper are done by using the Java applet for quiver mutations provided by Bernhard Keller, which is available at http://www.math.lsa.umich.edu/~fomin/cluster.html. This work is partially supported by JSPS KAKENHI Grant Number 18J13304 and the program for Leading Graduate School, MEXT, Japan. Publisher Copyright: © 2020, Institute of Mathematics. All rights reserved.

PY - 2020

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N2 - We give a method to compute presentations of saturated cluster modular groups. Using this, we obtain finite presentations of the saturated cluster modular groups of finite mutation type X6 and X7. We verify that the cluster modular groups of finite mutation type Ẽ6, Ẽ7, Ẽ8, G( ∗ , ∗ ) 2, X6 and X7 are virtually generated by cluster Dehn twists.

AB - We give a method to compute presentations of saturated cluster modular groups. Using this, we obtain finite presentations of the saturated cluster modular groups of finite mutation type X6 and X7. We verify that the cluster modular groups of finite mutation type Ẽ6, Ẽ7, Ẽ8, G( ∗ , ∗ ) 2, X6 and X7 are virtually generated by cluster Dehn twists.

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KW - Cluster modular groups

KW - Mapping class groups

KW - Quivers of finite mutation type

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Presentations of cluster modular groups and generation by cluster dehn twists

Abstract

Keywords

ASJC Scopus subject areas

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