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A Primer on Mapping Class Groups (PMS-49)$
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Benson Farb and Dan Margalit

Print publication date: 2011

Print ISBN-13: 9780691147949

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691147949.001.0001

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The Dehn-Nielsen-Baer Theorem

The Dehn-Nielsen-Baer Theorem

(p.219) Chapter Eight The Dehn-Nielsen-Baer Theorem
A Primer on Mapping Class Groups (PMS-49)

Benson Farb

Dan Margalit

Princeton University Press

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.

Keywords:   mapping class group, Dehn–Nielsen–Baer theorem, extended mapping class group, outer automorphism group, quasi-isometry, geometric group theory, homeomorphism, 3-manifold theory, harmonic maps

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