scholarly journals From Mapping Class Groups to Automorphism Groups of Free Groups

2005 ◽  
Vol 72 (2) ◽  
pp. 510-524 ◽  
Author(s):  
Nathalie Wahl
Keyword(s):  
Automorphism Groups ◽  
Free Groups ◽  
Mapping Class Groups ◽  
Mapping Class ◽  
Class Groups

An infinite family of braid group representations in automorphism groups of free groups

2020 ◽  
Vol 29 (10) ◽  
pp. 2042007
Author(s):  
Wonjun Chang ◽  
Byung Chun Kim ◽  
Yongjin Song
Keyword(s):  
Braid Group ◽  
Class Group ◽  
Automorphism Groups ◽  
Free Groups ◽  
Infinite Family ◽  
Braid Groups ◽  
Mapping Class ◽  
Group Representations ◽  
Class Groups ◽  
Branched Coverings

The [Formula: see text]-fold ([Formula: see text]) branched coverings on a disk give an infinite family of nongeometric embeddings of braid groups into mapping class groups. We, in this paper, give new explicit expressions of these braid group representations into automorphism groups of free groups in terms of the actions on the generators of free groups. We also give a systematic way of constructing and expressing these braid group representations in terms of a new gadget, called covering groupoid. We prove that each generator [Formula: see text] of braid group inside mapping class group induced by [Formula: see text]-fold covering is the product of [Formula: see text] Dehn twists on the surface.

Automorphisms of free groups and the mapping class groups of closed compact orientable surfaces

2007 ◽  
Vol 81 (1-2) ◽  
pp. 147-155
Author(s):  
S. I. Adyan ◽  
F. Grunewald ◽  
J. Mennicke ◽  
A. L. Talambutsa
Keyword(s):  
Free Groups ◽  
Mapping Class Groups ◽  
Mapping Class ◽  
Class Groups ◽  
Automorphisms Of Free Groups ◽  
Orientable 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
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