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Office Hours with a Geometric Group Theorist$
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Dan Margalit and Matt Clay

Print publication date: 2017

Print ISBN-13: 9780691158662

Published to Princeton Scholarship Online: May 2018

DOI: 10.23943/princeton/9780691158662.001.0001

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The Ping-Pong Lemma

The Ping-Pong Lemma

(p.85) Office Hour Five The Ping-Pong Lemma
Office Hours with a Geometric Group Theorist

Johanna Mangahas

Princeton University Press

This chapter considers an identifying feature of free groups: their ability to play ping-pong. In mathematics, you may encounter a group without immediately knowing which group it is. Fortunately, you can tell a group by how it acts. That is, a good group action (for example, action by isometries on a metric space) can reveal a lot about the group itself. This theme occupies a central place in geometric group theory. The ping-pong lemma, also dubbed Schottky lemma or Klein's criterion, gives a set of circumstances for identifying whether a group is a free group. The chapter first presents the statement, proof, and first examples using ping-pong before discussing ping-pong with Möbius transformations and hyperbolic geometry. Exercises and research projects are included.

Keywords:   free group, ping-pong, group action, geometric group theory, ping-pong lemma, Schottky lemma, Klein's criterion, Möbius transformation, hyperbolic geometry

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