## Dynamical forcing of circular groups

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**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.

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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