Tori, crystallographic groups and the realization of mapping classes

Lecture Notes in Mathematics - Finite Groups of Mapping Classes of Surfaces ◽

10.1007/bfb0090467 ◽

1981 ◽

pp. 65-139

Author(s):

Heiner Zieschang

Keyword(s):

Crystallographic Groups ◽

Mapping Classes

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Crystallographic Groups of Four-Dimensional Space.By H. Brown, R. Bülow, J. Neubüser, H. Wondratschek and H. Zassenhaus. Pp. XIV + 443. New York-Chichester-Brisbane-Toronto: John Wiley and Sons 1978. $38.50 (USA), £27.- (UK).

Zeitschrift für Kristallographie ◽

10.1524/zkri.1979.150.1-4.347 ◽

1979 ◽

Vol 150 (1-4) ◽

pp. 347-347

Author(s):

Werner Fischer

Keyword(s):

New York ◽

Crystallographic Groups

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Pseudo-Anosov Theory

10.23943/princeton/9780691147949.003.0015 ◽

2017 ◽

Cited By ~ 1

Author(s):

Benson Farb ◽

Dan Margalit

Keyword(s):

Braid Groups ◽

Intersection Numbers ◽

Branched Covers ◽

Algebraic Integers ◽

Dehn Twists ◽

Mapping Classes ◽

Dynamical Properties

This chapter focuses on the construction as well as the algebraic and dynamical properties of pseudo-Anosov homeomorphisms. It first presents five different constructions of pseudo-Anosov mapping classes: branched covers, constructions via Dehn twists, homological criterion, Kra's construction, and a construction for braid groups. It then proves a few fundamental facts concerning stretch factors of pseudo-Anosov homeomorphisms, focusing on the theorem that pseudo-Anosov stretch factors are algebraic integers. It also considers the spectrum of pseudo-Anosov stretch factors, along with the special properties of those measured foliations that are the stable (or unstable) foliations of some pseudo-Anosov homeomorphism. Finally, it describes the orbits of a pseudo-Anosov homeomorphism as well as lengths of curves and intersection numbers under iteration.

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