Outer automorphism groups of equivalence relations for mapping class group actions

2008 ◽  
Vol 78 (3) ◽  
pp. 622-638
Author(s):  
Yoshikata Kida
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Automorphism Groups ◽  
Group Actions ◽  
Outer Automorphism ◽  
Equivalence Relations ◽  
Mapping Class ◽  
Outer Automorphism Groups

The Dehn-Nielsen-Baer Theorem

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Automorphism Groups ◽  
Harmonic Maps ◽  
Outer Automorphism ◽  
Mapping Class ◽  
Original Proof ◽  
Outer Automorphism Groups ◽  
And Algebra ◽  
Manifold Theory

This chapter deals with the Dehn–Nielsen–Baer theorem, one of the most beautiful connections between topology and algebra in the mapping class group. It begins by defining the objects in the statement of the Dehn–Nielsen–Baer theorem, including the extended mapping class group and outer automorphism groups. It then considers the use of the notion of quasi-isometry in Dehn's original proof of the Dehn–Nielsen–Baer theorem. In particular, it discusses a theorem on the fundamental observation of geometric group theory, along with the property of being linked at infinity. It also presents the proof of the Dehn–Nielsen–Baer theorem and an analysis of the induced homeomorphism at infinity before concluding with two other proofs of the Dehn–Nielsen–Baer theorem, one inspired by 3-manifold theory and one using harmonic maps.

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 Automorphisms of braid groups on orientable surfaces

2016 ◽  
Vol 25 (05) ◽  
pp. 1650022
Author(s):  
Byung Hee An
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Automorphism Groups ◽  
Orientable Surface ◽  
Braid Groups ◽  
Mapping Class ◽  
Characteristic Subgroup ◽  
Group Extensions ◽  
Extended Mapping ◽  
Orientable Surfaces

In this paper, we compute the automorphism groups [Formula: see text] and [Formula: see text] of braid groups [Formula: see text] and [Formula: see text] on every orientable surface [Formula: see text], which are isomorphic to group extensions of the extended mapping class group [Formula: see text] by the transvection subgroup except for a few cases. We also prove that [Formula: see text] is always a characteristic subgroup of [Formula: see text], unless [Formula: see text] is a twice-punctured sphere and [Formula: see text].

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
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