## 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.

### MSC:

 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