scholarly journals Mapping class group actions from Hopf monoids and ribbon graphs

2021 ◽  
Author(s):  
Catherine Meusburger ◽  
Thomas Voß
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Group Actions ◽  
Mapping Class ◽  
Ribbon Graphs

scholarly journals Rigidity of mapping class group actions on S1

2020 ◽  
Vol 24 (3) ◽  
pp. 1211-1223
Author(s):  
Kathryn Mann ◽  
Maxime Wolff
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Group Actions ◽  
Mapping Class

scholarly journals Graphs of curves on infinite-type surfaces with mapping class group actions

2018 ◽  
Vol 68 (6) ◽  
pp. 2581-2612 ◽  
Author(s):  
Matthew Gentry Durham ◽  
Federica Fanoni ◽  
Nicholas G. Vlamis
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Group Actions ◽  
Mapping Class ◽  
Infinite Type

Generating the Mapping Class Group

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Exact Sequence ◽  
Simplicial Complex ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Detailed Structure ◽  
Inductive Step ◽  
Closed Curves ◽  
Combinatorial Object ◽  
Generating Sets

This chapter considers the Dehn–Lickorish theorem, which states that when g is greater than or equal to 0, the mapping class group Mod(Sɡ) is generated by finitely many Dehn twists about nonseparating simple closed curves. The theorem is proved by induction on genus, and the Birman exact sequence is introduced as the key step for the induction. The key to the inductive step is to prove that the complex of curves C(Sɡ) is connected when g is greater than or equal to 2. The simplicial complex C(Sɡ) is a useful combinatorial object that encodes intersection patterns of simple closed curves in Sɡ. More detailed structure of C(Sɡ) is then used to find various explicit generating sets for Mod(Sɡ), including those due to Lickorish and to Humphries.

scholarly journals The local-to-global property for Morse quasi-geodesics

2021 ◽  
Author(s):  
Jacob Russell ◽  
Davide Spriano ◽  
Hung Cong Tran
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Type Theorem ◽  
Hyperbolic Groups ◽  
Global Property ◽  
Mapping Class ◽  
Relatively Hyperbolic Groups ◽  
Convex Cocompact ◽  
Relatively Hyperbolic ◽  
Translation Length

AbstractWe show the mapping class group, $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgroups of these groups. In the case of the mapping class group, this gives combination theorems for convex cocompact subgroups. We show a number of additional consequences of this local-to-global property, including a Cartan–Hadamard type theorem for detecting hyperbolicity locally and discreteness of translation length of conjugacy classes of Morse elements with a fixed gauge. To prove the relatively hyperbolic case, we develop a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces, extending work of Hruska.

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