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Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists
Symmetry, integrability and geometry: methods and applications, Tome 16 (2020) Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $X_6$ and $X_7$. We verify that the cluster modular groups of finite mutation type $\widetilde{E}_6$, $\widetilde{E}_7$, $\widetilde{E}_8$, $G_2^{(*,*)}$, $X_6$ and $X_7$ are virtually generated by cluster Dehn twists.
Keywords: cluster algebras, cluster modular groups, mapping class groups, quivers of finite mutation type. 
@article{SIGMA_2020_16_a24,
     author = {Tsukasa Ishibashi},
     title = {Presentations of {Cluster} {Modular} {Groups} and {Generation} by {Cluster} {Dehn} {Twists}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2020},
     volume = {16},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a24/}
}
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Tsukasa Ishibashi. Presentations of Cluster Modular Groups and Generation by Cluster Dehn Twists. Symmetry, integrability and geometry: methods and applications, Tome 16 (2020). http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a24/

[1] Assem I., Schiffler R., Shramchenko V., “Cluster automorphisms”, Proc. Lond. Math. Soc., 104 (2012), 1271–1302, arXiv: 1009.0742 | DOI | MR | Zbl

[2] Bridgeland T., Smith I., “Quadratic differentials as stability conditions”, Publ. Math. Inst. Hautes Études Sci., 121 (2015), 155–278, arXiv: 1302.7030 | DOI | MR | Zbl

[3] Brown K.S., “Presentations for groups acting on simply-connected complexes”, J. Pure Appl. Algebra, 32 (1984), 1–10 | DOI | MR | Zbl

[4] Farb B., Margalit D., A primer on mapping class groups, Princeton Mathematical Series, 49, Princeton University Press, Princeton, NJ, 2012 | DOI | MR

[5] Felikson A., Shapiro M., Tumarkin P., “Cluster algebras of finite mutation type via unfoldings”, Int. Math. Res. Not., 2012 (2012), 1768–1804, arXiv: 1006.4276 | DOI | MR | Zbl

[6] Felikson A., Shapiro M., Tumarkin P., “Skew-symmetric cluster algebras of finite mutation type”, J. Eur. Math. Soc., 14 (2012), 1135–1180, arXiv: 0811.1703 | DOI | MR | Zbl

[7] Fock V.V., Goncharov A.B., “Cluster ${\mathcal X}$-varieties, amalgamation, and Poisson–Lie groups”, Algebraic Geometry and Number Theory, Progr. Math., 253, Birkhäuser Boston, Boston, MA, 2006, 27–68, arXiv: math.RT/0508408 | DOI | MR | Zbl

[8] Fock V.V., Goncharov A.B., “Moduli spaces of local systems and higher Teichmüller theory”, Publ. Math. Inst. Hautes Études Sci., 2006, 1–211, arXiv: math.AG/0311149 | DOI | MR | Zbl

[9] Fock V.V., Goncharov A.B., “Cluster ensembles, quantization and the dilogarithm”, Ann. Sci. Éc. Norm. Supér. (4), 42 (2009), 865–930, arXiv: math.AG/0311245 | DOI | MR | Zbl

[10] Fock V.V., Goncharov A.B., “The quantum dilogarithm and representations of quantum cluster varieties”, Invent. Math., 175 (2009), 223–286, arXiv: math.QA/0702397 | DOI | MR | Zbl

[11] Fomin S., Shapiro M., Thurston D., “Cluster algebras and triangulated surfaces. I Cluster complexes”, Acta Math., 201 (2008), 83–146, arXiv: math.RA/0608367 | DOI | MR | Zbl

[12] Fomin S., Zelevinsky A., “Cluster algebras. II Finite type classification”, Invent. Math., 154 (2003), 63–121, arXiv: math.RA/0208229 | DOI | MR | Zbl

[13] Fomin S., Zelevinsky A., “Cluster algebras. IV Coefficients”, Compos. Math., 143 (2007), 112–164, arXiv: math.RA/0602259 | DOI | MR | Zbl

[14] Fraser C., “Quasi-homomorphisms of cluster algebras”, Adv. in Appl. Math., 81 (2016), 40–77, arXiv: 1509.05385 | DOI | MR | Zbl

[15] Ishibashi T., “On a Nielsen–Thurston classification theory for cluster modular groups”, Ann. Inst. Fourier (Grenoble), 69 (2019), 515–560, arXiv: 1704.06586 | DOI | MR | Zbl

[16] Kato A., Terashima Y., “Quiver mutation loops and partition $q$-series”, Comm. Math. Phys., 336 (2015), 811–830, arXiv: 1403.6569 | DOI | MR | Zbl

[17] Kim H.K., Yamazaki M., “Comments on exchange graphs in cluster algebras”, Exp. Math., 29 (2020), 79–100, arXiv: 1612.00145 | DOI | MR | Zbl

[18] Penner R.C., Decorated Teichmüller theory, QGM Master Class Series, European Mathematical Society (EMS), Zürich, 2012 | DOI | MR

[19] Zickert C.K., “Fock–Goncharov coordinates for rank two Lie groups”, Math. Z., 294 (2020), 251–286, arXiv: 1605.08297 | DOI | MR | Zbl