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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Dynamical forcing of circular groups
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by Danny Calegari PDF
Trans. Amer. Math. Soc. 358 (2006), 3473-3491 Request permission

Abstract:

In this paper we introduce and study the notion of dynamical forcing. Basically, we develop a toolkit of techniques to produce finitely presented groups which can only act on the circle with certain prescribed dynamical properties. As an application, we show that the set $X \subset \mathbb {R}/\mathbb {Z}$ consisting of rotation numbers $\theta$ which can be forced by finitely presented groups is an infinitely generated $\mathbb {Q}$–module, containing countably infinitely many algebraically independent transcendental numbers. Here a rotation number $\theta$ is forced by a pair $(G_\theta ,\alpha )$, where $G_\theta$ is a finitely presented group $G_\theta$ and $\alpha \in G_\theta$ is some element, if the set of rotation numbers of $\rho (\alpha )$ as $\rho$ varies over $\rho \in \operatorname {Hom}(G_\theta ,\operatorname {Homeo}^+(S^1))$ is precisely the set $\lbrace 0, \pm \theta \rbrace$. We show that the set of subsets of $\mathbb {R}/\mathbb {Z}$ which are of the form \[ \operatorname {rot}(X(G,\alpha )) = \lbrace r \in \mathbb {R}/\mathbb {Z} \; | \; r = \operatorname {rot}(\rho (\alpha )), \rho \in \operatorname {Hom}(G,\operatorname {Homeo}^+(S^1)) \rbrace ,\] where $G$ varies over countable groups, are exactly the set of closed subsets which contain $0$ and are invariant under $x \to -x$. Moreover, we show that every such subset can be approximated from above by $\operatorname {rot}(X(G_i,\alpha _i))$ for finitely presented $G_i$. As another application, we construct a finitely generated group $\Gamma$ which acts faithfully on the circle, but which does not admit any faithful $C^1$ action, thus answering in the negative a question of John Franks.
References
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Additional Information
  • Danny Calegari
  • Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125
  • MR Author ID: 605373
  • Email: dannyc@its.caltech.edu
  • Received by editor(s): December 8, 2003
  • Received by editor(s) in revised form: May 24, 2004
  • Published electronically: June 10, 2005
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 358 (2006), 3473-3491
  • MSC (2000): Primary 58D05; Secondary 57S99
  • DOI: https://doi.org/10.1090/S0002-9947-05-03754-2
  • MathSciNet review: 2218985