Iteration of mapping classes on a Bers slice: examples of algebraic and geometric limits of hyperbolic 3-manifolds

1997 ◽  
pp. 81-106 ◽  
Author(s):  
Jeffrey F. Brock
Keyword(s):  
Mapping Classes

Pseudo-Anosov Theory

2017 ◽  
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.

scholarly journals Commuting conjugates of finite-order mapping classes

2020 ◽  
Vol 209 (1) ◽  
pp. 69-93
Author(s):  
Neeraj K. Dhanwani ◽  
Kashyap Rajeevsarathy
Keyword(s):  
Finite Order ◽  
Mapping Classes

scholarly journals The stratum of random mapping classes

2017 ◽  
Vol 38 (7) ◽  
pp. 2666-2682 ◽  
Author(s):  
VAIBHAV GADRE ◽  
JOSEPH MAHER
Keyword(s):  
Random Walks ◽  
Mapping Class Group ◽  
Class Group ◽  
Sample Path ◽  
Mapping Class ◽  
Elementary Subgroup ◽  
Random Mapping ◽  
Mapping Classes ◽  
First Moment

We consider random walks on the mapping class group that have finite first moment with respect to the word metric, whose support generates a non-elementary subgroup and contains a pseudo-Anosov map whose invariant Teichmüller geodesic is in the principal stratum. For such random walks, we show that mapping classes along almost every infinite sample path are eventually pseudo-Anosov, with invariant Teichmüller geodesics in the principal stratum. This provides an answer to a question of Kapovich and Pfaff [Internat. J. Algebra Comput.25, 2015 (5) 745–798].

scholarly journals Examples of Non-trivial Contact Mapping Classes in all Dimensions

2016 ◽  
Vol 2016 (15) ◽  
pp. 4784-4806 ◽  
Author(s):  
Patrick Massot ◽  
Niederkrüger Klaus
Keyword(s):  
Contact Mapping ◽  
Mapping Classes

scholarly journals A zoo of growth functions of mapping class sets

2018 ◽  
Vol 12 (03) ◽  
pp. 841-855 ◽  
Author(s):  
Fedor Manin
Keyword(s):  
Mapping Class ◽  
Rational Homotopy ◽  
Growth Functions ◽  
Lipschitz Maps ◽  
Rational Cohomology ◽  
Rational Homotopy Type ◽  
Number Formula ◽  
Mapping Classes ◽  
Simply Connected ◽  
The Universe

Suppose [Formula: see text] and [Formula: see text] are finite complexes, with [Formula: see text] simply connected. Gromov conjectured that the number of mapping classes in [Formula: see text] which can be realized by [Formula: see text]-Lipschitz maps grows asymptotically as [Formula: see text], where [Formula: see text] is an integer determined by the rational homotopy type of [Formula: see text] and the rational cohomology of [Formula: see text]. This conjecture was disproved in a recent paper of the author and Weinberger; we gave an example where the “predicted” growth is [Formula: see text] but the true growth is [Formula: see text]. Here we show, via a different mechanism, that the universe of possible such growth functions is quite large. In particular, for every rational number [Formula: see text], there is a pair [Formula: see text] for which the growth of [Formula: see text] is essentially [Formula: see text].

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