scholarly journals Cluster Automorphisms and the Marked Exchange Graphs of Skew-Symmetrizable Cluster Algebras

10.37236/6230 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
John W Lawson
Keyword(s):  
Mapping Class Groups ◽  
Cluster Algebras ◽  
Mapping Class ◽  
Symmetric Matrices ◽  
Class Groups ◽  
Exchange Graph ◽  
Symmetrizable Matrices

Cluster automorphisms have been shown to have links to the mapping class groups of surfaces, maximal green sequences and to exchange graph automorphisms for skew-symmetric cluster algebras. In this paper we generalise these results to the skew-symmetrizable case by introducing a marking on the exchange graph. Many skew-symmetrizable matrices unfold to skew-symmetric matrices and we consider how cluster automorphisms behave under this unfolding with applications to coverings of orbifolds by surfaces.

The Nielsen-Thurston Classification

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Hyperbolic Geometry ◽  
Mapping Class Groups ◽  
Classification Theorem ◽  
Mapping Class ◽  
Fundamental Result ◽  
Class Groups ◽  
Mapping Torus ◽  
Higher Genus ◽  
Manifold Theory

This chapter explains and proves the Nielsen–Thurston classification of elements of Mod(S), one of the central theorems in the study of mapping class groups. It first considers the classification of elements for the torus of Mod(T² before discussing higher-genus analogues for each of the three types of elements of Mod(T². It then states the Nielsen–Thurston classification theorem in various forms, as well as a connection to 3-manifold theory, along with Thurston's geometric classification of mapping torus. The rest of the chapter is devoted to Bers' proof of the Nielsen–Thurston classification. The collar lemma is highlighted as a new ingredient, as it is also a fundamental result in the hyperbolic geometry of surfaces.

On the Teichmüller tower of mapping class groups

2000 ◽  
Vol 2000 (521) ◽  
pp. 1-24 ◽  
Author(s):  
Allen Hatcher ◽  
Pierre Lochak ◽  
Leila Schneps
Keyword(s):  
Mapping Class Groups ◽  
Mapping Class ◽  
Class Groups

scholarly journals Right-angled Artin groups as normal subgroups of mapping class groups

2021 ◽  
Vol 157 (8) ◽  
pp. 1807-1852
Author(s):  
Matt Clay ◽  
Johanna Mangahas ◽  
Dan Margalit
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class Groups ◽  
Braid Groups ◽  
Mapping Class ◽  
Artin Groups ◽  
Normal Subgroups ◽  
Class Groups ◽  
Right Angled Artin Groups ◽  
Proper Normal Subgroup

We construct the first examples of normal subgroups of mapping class groups that are isomorphic to non-free right-angled Artin groups. Our construction also gives normal, non-free right-angled Artin subgroups of other groups, such as braid groups and pure braid groups, as well as many subgroups of the mapping class group, such as the Torelli subgroup. Our work recovers and generalizes the seminal result of Dahmani–Guirardel–Osin, which gives free, purely pseudo-Anosov normal subgroups of mapping class groups. We give two applications of our methods: (1) we produce an explicit proper normal subgroup of the mapping class group that is not contained in any level $m$ congruence subgroup and (2) we produce an explicit example of a pseudo-Anosov mapping class with the property that all of its even powers have free normal closure and its odd powers normally generate the entire mapping class group. The technical theorem at the heart of our work is a new version of the windmill apparatus of Dahmani–Guirardel–Osin, which is tailored to the setting of group actions on the projection complexes of Bestvina–Bromberg–Fujiwara.

scholarly journals Embeddings and the (virtual) cohomological dimension of the braid and mapping class groups of surfaces

2018 ◽  
Vol 10 (1) ◽  
pp. 41-61
Author(s):  
Daciberg Lima Gonçalves ◽  
John Guaschi ◽  
Miguel Maldonado
Keyword(s):  
Cohomological Dimension ◽  
Mapping Class Groups ◽  
Mapping Class ◽  
Class Groups ◽  
Virtual Cohomological Dimension

scholarly journals Presentations for the punctured mapping class groups in terms of Artin groups

2001 ◽  
Vol 1 (1) ◽  
pp. 73-114 ◽  
Author(s):  
Catherine Labruere ◽  
Luis Paris
Keyword(s):  
Mapping Class Groups ◽  
Mapping Class ◽  
Artin Groups ◽  
Class Groups

scholarly journals On crossed homomorphisms on symplectic mapping class groups

2008 ◽  
Vol 360 (09) ◽  
pp. 4815-4839 ◽  
Author(s):  
Ryoji Kasagawa
Keyword(s):  
Mapping Class Groups ◽  
Mapping Class ◽  
Class Groups ◽  
Symplectic Mapping

L-algebras and topology

2021 ◽  
Author(s):  
Wolfgang Rump
Keyword(s):  
Mapping Class Groups ◽  
Unitary Groups ◽  
Mapping Class ◽  
Heyting Algebras ◽  
Baxter Equation ◽  
Prime Spectrum ◽  
Class Groups ◽  
Orthomodular Lattices ◽  
And Topology ◽  
Torsion Free

[Formula: see text]-algebras are based on an equation which is fundamental in the construction of various torsion-free groups, including spherical Artin groups, Riesz groups, certain mapping class groups, para-unitary groups, and structure groups of set-theoretic solutions to the Yang–Baxter equation. A topological study of [Formula: see text]-algebras is initiated. A prime spectrum is associated to certain (possibly all) [Formula: see text]-algebras, including three classes of [Formula: see text]-algebras where the ideals are determined in a more explicite fashion. Known results on orthomodular lattices, Heyting algebras, or quantales are extended and revisited from an [Formula: see text]-algebraic perspective.

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