Groups acting on the circle. (English) Zbl 1044.37033

The paper is a beautiful survey of the recent progress about groups of homeomorphisms of the circle \(S^1\). It is very well written, essentially self-contained, includes many proofs and gives an extensive list of references.
Section \(3\) serves as an introduction to the subject and discusses the standard action on \(S^1\) of the group \(PSL (2, \mathbb R)\) and piecewise linear groups including the Thompson group. Section \(4\) discusses the basic topological and group theoretical properties of the group \(\text{Homeo}_+ (S^1)\) of orientation-preserving homeomorphisms of \(S^1\). It is shown that for a generic set of pairs \((f,g)\) of elements of \(\text{Homeo}_+ (S^1)\) the group generated by \((f,g)\) is a free group in two generators. Transitive actions of Lie groups on \(S^1\) are classified.
In Section \(5\), the rotation number of an element of \(\text{Homeo}_+ (S^1)\) is introduced. It is shown that the action of a homeomorphism with an exceptional minimal set is semi-conjugate to an action with dense orbits. Moreover, it is shown that \(\text{Homeo}_+ (S^1)\) is a simple group. This leads to a version of Tits alternative for subgroups of \(\text{Homeo}_+ (S^1)\).
Section \(6\) is devoted to cohomological properties of subgroups \(\Gamma\) of \(\text{Homeo}_+ (S^1)\). Namely, for every such group the restriction of the Euler class is an element of \(H^2 (\Gamma; \mathbb Z)\). The classical Milnor-Wood inequality gives a bound on this class in the case that \(\Gamma\) is isomorphic to the fundamental group of a closed surface. More importantly, the bounded Euler class in the second bounded cohomology group \(H^2 _b (\Gamma ; \mathbb R)\) contains the usual Euler class as well as infomations about rotation numbers.
Section \(7\) contains recent results about actions on \(S^1\) of lattices in Lie groups of higher rank. It begins with the discussion of the result of Witte that for \(n \geq 3\), any homeomorphism \(\varphi \colon SL (n, \mathbb Z) \to \text{Homeo}_+ (S^1)\) has finite image. This result was extended by the author in [Invent. Math. 137, 199–231 (1999; Zbl 0995.57006)] as follows: Let \(\Gamma\) be a lattice in a simple Lie group \(G\) with real rank \(\geq 2\). Then any action of \(\Gamma\) on \(S^1\) has a finite orbit. A sketch of the proof of this result is presented. The last part of the paper contains a discussion of examples.


37E10 Dynamical systems involving maps of the circle
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
57S99 Topological transformation groups
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory


Zbl 0995.57006