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Dynamical forcing of circular groups

Calegari, Danny (2006) Dynamical forcing of circular groups. Transactions of the American Mathematical Society, 358 (8). pp. 3473-3491. ISSN 0002-9947. https://resolver.caltech.edu/CaltechAUTHORS:20110120-094122587

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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 ⊂ R/Z consisting of rotation numbers θ which can be forced by finitely presented groups is an infinitely generated Q-module, containing countably infinitely many algebraically independent transcendental numbers. Here a rotation number θ is forced by a pair (G_θ, α), where G_θ is a finitely presented group G_θ and α ∈ G_θ is some element, if the set of rotation numbers of ρ(α) as ρ ∈ Hom(G_θ, Homeo^(+)(S^1)) is precisely the set {0,±θ}. We show that the set of subsets of R/Z which are of the form rot(X(G, α)) = {r ∈ R/Z | r = rot(ρ(α)), ρ ∈ Hom(G, Homeo^(+)(S^1))}, where G varies over countable groups, are exactly the set of closed subsets which contain 0 and are invariant under x→−x. Moreover, we show that every such subset can be approximated from above by rot(X(G_i, α_i)) for finitely presented G_i. As another application, we construct a finitely generated group Γ 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.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1090/S0002-9947-05-03754-2 DOIUNSPECIFIED
http://www.ams.org/journals/tran/2006-358-08/S0002-9947-05-03754-2/home.htmlPublisherUNSPECIFIED
Additional Information:© 2005 American Mathematical Society. Reverts to public domain 28 years from publication. Received by the editors December 8, 2003 and, in revised form, May 24, 2004; Posted: June 10, 2005. The subject matter in this paper was partly inspired by a discussion with John Franks and Amie Wilkinson, and a comment in an email from Étienne Ghys. I thank Nathan Dunfield for some excellent comments on an earlier version of this paper, and Hee Oh for some useful info about arithmetic lattices. Thanks as well to the referee, for catching a number of errors, especially in some of the formulae.
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Other Numbering System NameOther Numbering System ID
MathSciNet review2218985
Issue or Number:8
Classification Code:2000 Mathematics Subject Classification: Primary 58D05; Secondary 57S99.
Record Number:CaltechAUTHORS:20110120-094122587
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20110120-094122587
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:21823
Collection:CaltechAUTHORS
Deposited By: Jason Perez
Deposited On:20 Jan 2011 23:51
Last Modified:03 Oct 2019 02:30

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