Steinberg representations and duality properties of arithmetic groups, mapping class groups, and outer automorphism groups of free groups

2009 ◽  
pp. 71-98
Author(s):  
Lizhen Ji
Keyword(s):  
Automorphism Groups ◽  
Free Groups ◽  
Outer Automorphism ◽  
Mapping Class Groups ◽  
Mapping Class ◽  
Arithmetic Groups ◽  
Class Groups ◽  
Outer Automorphism 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.

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.

Arithmetic Groups vs. Mapping Class Groups: Similarities, Analogies and Differences

2011 ◽  
pp. 1637-1708 ◽  
Author(s):  
Benson Farb ◽  
Lizhen Ji ◽  
Enrico Leuzinger ◽  
Werner Müller
Keyword(s):  
Mapping Class Groups ◽  
Mapping Class ◽  
Arithmetic Groups ◽  
Class Groups
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