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Transactions of the American Mathematical Society

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Dynamical forcing of circular groups

Author: Danny Calegari
Journal: Trans. Amer. Math. Soc. 358 (2006), 3473-3491
MSC (2000): Primary 58D05; Secondary 57S99
Published electronically: June 10, 2005
MathSciNet review: 2218985
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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 \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

\begin{displaymath}\operatorname{rot}(X(G,\alpha)) = \lbrace r \in \mathbb{R} /... ... \in \operatorname{Hom}(G,\operatorname{Homeo}^+(S^1)) \rbrace,\end{displaymath}

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.

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Additional Information

Danny Calegari
Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125

Received by editor(s): December 8, 2003
Received by editor(s) in revised form: May 24, 2004
Published electronically: June 10, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.