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Classification of the Group Actions on the Real Line and Circle

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Classification of the Group Actions on the Real Line and Circle Algebra i analiz St. Petersburg Math. J. Tom 19 (2007), 2 Vol. 19 (2008), No. 2, Pages 279–296 S 1061-0022(08)00999-0 Article electronically published on February 7, 2008 CLASSIFICATION OF THE GROUP ACTIONS ON THE REAL LINE AND CIRCLE A. V. MALYUTIN Abstract. The group actions on the real line and circle are classified. It is proved that each minimal continuous action of a group on the circle is either a conjugate of an isometric action, or a finite cover of a proximal action. It is also shown that each minimal continuous action of a group on the real line either is conjugate to an isometric action, or is a proximal action, or is a cover of a proximal action on the circle. As a corollary, it is proved that a continuous action of a group on the circle either has a finite orbit, or is semiconjugate to a minimal action on the circle that is either isometric or proximal. As a consequence, a new proof of the Ghys–Margulis alternative is obtained. Introduction In this paper, by an action of a group on a topological space we mean an action by homeomorphisms; by (semi)conjugacy of actions we mean topological (semi)conjugacy. An action of a group G on a Hausdorff topological space X is said to be proximal if for any two points x and y in X there exists a sequence {gk} in G such that the sequences {gkx} and {gky} converge to one and the same point. The key result of the paper is the following theorem (see §1 for the definitions involved). Theorem 1. Let X be either the real line (R) or the circle (S1).Letφ : G → Homeo(X) be a minimal action of a group G on X (by homeomorphisms). Then exactly one of the following three assertions is true: (i) the action φ is conjugate to an isometric action; (ii) the action φ is proximal; (iii) the action φ is a nontrivial cover of a minimal and proximal action on the circle; i.e., φ is semiconjugate to a minimal and proximal action on the circle via a nontrivial covering map. If X = S1, then assertions (ii) and (iii) can be united, and Theorem 1 takes the following form. Theorem 1 (Case of S1). Every minimal continuous action of a group on the circle is either a conjugate of an isometric action or a finite cover of a proximal action. Theorem 1 implies the following classification (see §7 for the proof). 2000 Mathematics Subject Classification. Primary 54H15; Secondary 57S25, 57M60, 54H20, 37E05, 37E10. Key words and phrases. Circle, line, group of homeomorphisms, action, proximal, distal, semi- conjugacy. The author was partially supported by grant NSh-4329.2006.1 and by RFBR grant no. 05-01-00899. c 2008 American Mathematical Society 279 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 280 A. V. MALYUTIN Corollary 2 (S1). Let φ : G → Homeo(S1) be an action of a group G on the circle (by homeomorphisms). Then exactly one of the following three conditions is fulfilled: (a) the action φ has a finite orbit; (b) the action φ is semiconjugate to an isometric minimal action on the circle via a continuous locally monotone semiconjugacy of degree 1; (c) the action φ is semiconjugate to a minimal and proximal action on the circle via a continuous locally monotone semiconjugacy. Theorem 1 also implies a similar classification (which, however, includes more cases) for the actions of groups on the real line. A remark about alternatives. It turns out that if an action φ : G → Homeo(S1) meets either condition (a) or condition (b) of Corollary 2, then S1 bears a G-invariant Borel probability measure (this is explained in the proof of Corollary 2 in §7). On the other hand, it can be shown that each continuous proximal action of a group on the circle is strongly proximal in the sense of Furstenberg (see [3]; an action is strongly proximal if and only if the induced action on the associated space of Borel probability measures is proximal). Therefore, Corollary 2 implies the following alternative (where the cases are not mutually exclusive). Every action φ : G → Homeo(S1) either admits an invariant Borel probability measure, or is semiconjugate to a strongly proximal action on the circle. It is easy to show that if an action of a group G on the circle is minimal and proximal, then G contains a free non-Abelian subgroup (see Proposition 6 and Remark 6.2 for the proof). Therefore, Corollary 2 implies the Ghys alternative, which was proved by G. A. Margulis in [6]: If a group G acts on the circle S1 by homeomorphisms, then either there exists a G-invariant Borel probability measure on S1,orG contains a free non-Abelian subgroup. A remark about centralizers. Let X be either the real line or the circle, and let φ : G → Homeo(X) be a minimal action (as in Theorem 1). Let Z denote the centralizer of the subgroup φ(G) ∩ Homeo+(X) in the group Homeo(X). Assertions (i), (ii), and (iii) of Theorem 1 can be reformulated in terms of the central- izer Z as follows: • (i) ⇐⇒ Z is isomorphic to the Abelian group X; • (ii) ⇐⇒ Z is trivial; • (iii) ⇐⇒ Z is a cyclic group (infinite if X = R, and finite if X = S1). We also remark that in case (iii) the orbit space X/Z is homeomorphic to the circle, the corresponding projection p : X → X/Z is a nontrivial covering (universal if X = R, and finite-sheeted if X = S1), and there exists a minimal and proximal action ψ : G → Homeo(X/Z) such that the covering p is a semiconjugacy between φ and ψ. On uniqueness. Let G be a group, and let X ∈{R, S1}. Suppose that an action 1 φ : G → Homeo(X) is semiconjugate to two actions ψi : G → Homeo(S )(i =1, 2) 1 via continuous locally monotone semiconjugacies pi : X → S (i =1, 2), respectively. Suppose moreover that both ψ1 and ψ2 are minimal and proximal. Then it can be shown 1 1 that ψ1 and ψ2 are conjugate. Furthermore, there exists a conjugacy q : S → S between ψ1 and ψ2 such that p2 = q ◦ p1. It follows, in particular, that if an action φ : G → Homeo(S1) meets condition (c) of Corollary 2, then the absolute value of the degree of the semiconjugacy mentioned in condition (c) is an invariant of φ. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use CLASSIFICATION OF THE GROUP ACTIONS ON THE REAL LINE AND CIRCLE 281 Structure of the paper. §1 contains the basic definitions. In §2 we prove Proposition 2, which says that each distal action of a group on X ∈ {R, S1} is conjugate to an isometric action. §3 contains several lemmas involved in the proof of Proposition 4 (see §4). In §4 we prove Proposition 4, which concerns minimal group actions on the real line that are neither proximal nor distal. In §5, we show that each minimal and nondistal action of a group on the circle is a cover of a minimal and proximal action (see Proposition 5). In §6, we prove some properties of a minimal and proximal group action on the circle. In §7, Theorem 1 and Corollary 2 are deduced from the results of the preceding sections. §1. Preliminaries 1.1. Group actions on topological spaces. Let G be a group, X a topological space. An action of G on X is a homomorphism φ from G to the group Homeo(X)ofall homeomorphisms of X: φ : G → Homeo(X). The image of a point x ∈ X under a homeomorphism φ(g)(g ∈ G) is denoted by g(x) or gx.ForF ⊂ G and A ⊂ X, we define F (A):={g(x) | g ∈ F, x ∈ A}. For a point x ∈ X,theorbit of x is the set G(x). A subset A ⊂ X is said to be G-invariant if g(A)=A for each g ∈ G. If H is a subgroup of Homeo(X), then by the action of H on X we mean the inclusion H → Homeo(X). For a given action φ : G → Homeo(X) and a subgroup G of G,bythe action of G on X we mean the restriction φ|G : G → Homeo(X). An action of a group on a topological space is said to be minimal if each orbit of the action is dense (or, equivalently, if the action has no proper closed invariant subsets). Two actions φ : G → Homeo(X)andψ : G → Homeo(Y )areconjugate if there exists a homeomorphism f : X → Y such that g(f(x)) = f(g(x)) for any x ∈ X and any g ∈ G. In this case, we also say that φ is conjugate to ψ via f,andthatf is a conjugacy between φ and ψ. An action φ : G → Homeo(X)issemiconjugate to an action ψ : G → Homeo(Y )if there exists a continuous surjection f : X → Y such that g(f(x)) = f(g(x)) for any x ∈ X and any g ∈ G. In this case, we also say that φ is semiconjugate to ψ via f,andthatf is a semiconjugacy between φ and ψ. If an action φ is semiconjugate to an action ψ via a covering map, we say that φ is a cover of ψ. 1.2. Proximal and distal actions. Let a group G act on a Hausdorff topological space X.
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