Periodic points for amenable group actions on uniquely arcwise connected continua

We show that if $G$ is a countable amenable group acting on a uniquely arcwise connected continuum $X$, then $G$ has either a fixed point or a 2-periodic point in $X$.


Basic notions
Let X be a topological space and let Homeo(X) be the homeomorphism group of X. Let G be a group. A group homomorphism φ : G → Homeo(X) is called an action of G on X. For brevity, we usually use gx or g(x) instead of φ(g)(x) for g ∈ G and x ∈ X. The orbit of x ∈ X under the action of G is the set Gx ≡ {gx : g ∈ G}; x is called a periodic point of G if Gx is finite, and the number of all elements in Gx is called the order of x; if x is a periodic point of order n for some positive integer n, then x is called an n-periodic point; if x is of order 1, then x is called a fixed point of G, that is gx = x for all g ∈ G. A subset Y of X is called G-invariant, if g(Y ) ⊂ Y for all g ∈ G. A Borel measure µ on X is called G-invariant if µ(g(A)) = µ(A) for every Borel set A in X and every g ∈ G.
Amenability was first introduced by von Neumann. Recall that a countable group G is called an amenable group if there is a sequence of finite sets F i (i = 1, 2, 3, . . . ) such that lim i→∞ |gF i △F i | |F i | = 0 for every g ∈ G, where |F i | is the number of elements in F i ; the set F i is called a Følner set. It is well known that solvable groups and finite groups are amenable; every subgroup of an amenable group is amenable. It is also known that any group containing a free noncommutative subgroup is not amenable. An important characterization of countable amenable group is that G is amenable if and only if every action of G on a compact metric space X has a G-invariant Borel probability measure on X. One may consult [8] for a systematic introduction to amenability. By a continuum, we mean a connected compact metric space. A continuum is nondegenerate if it is not a single point. An arc is a continuum which is homeomorphic to the closed interval [0, 1]. A continuum X is uniquely arcwise connected if for any two points x = y ∈ X there is a unique arc [x, y] in X, which connects x and y. A dendrite is a locally connected, uniquely arcwise connected continuum. A tree is a dendrite which is the union of finitely many arcs. Clearly, the class of uniquely arcwise connected continua is strictly larger than that of dendrites. For example, the Warsaw circle is uniquely arcwise connected but not locally connected.

Backgrounds and the main theorem
For an action of a group G on a topological space X, an interesting question is whether there exists a fixed point or a periodic point of G in X. The answer to this question depends on the topology of X and the algebraic structure of G.
In 1975, Mohler proved in [6] that every homeomorphism (i.e., Z-action) on a uniquely arcwise connected continuum has a fixed point, which answered a question proposed by Bing (see [1]). In 2009, this result is generalized to nilpotent group actions by Shi and Sun (see [9]). In 2010, Shi and Zhou further showed that every solvable group action on such continua has either a fixed point or a 2-periodic point (see [11]). 1n 2016, Shi and Ye proved that every countable amenable group action on a dendrite either has a fixed point or has a 2-periodic point (see [10]). One may consult [2,4,5] for some interesting discussions about fixed point theory for mappings on uniquely arcwise connected continua. We also remak that a continuous map on a uniquely arcwise connected continuum may have no fixed points (see [12]).
We get the following theorem in this paper, which generalize all the corresponding results stated above. Theorem 1.2. Let G be a countable amenable group and let X be a uniquely arcwise connected continuum. Suppose φ : G → Homeo(X) is a group action. Then G has either a fixed point or a 2-periodic point in X.
In Section 2, we introduce some basic notions and results concerning the structure and mapping properties of uniquely arcwise connected continua. In Section 3, we construct a convex metric on a special class of arcwise connected subsets of uniquely arcwise connected continua and study their completions with respect to this metric. Specially, we establish a connection between the group actions on dendrites and that on uniquely arcwise connected continua. Based on the connection established in Section 3 and the main theorem in [10], we prove Theorem 1.2 in Section 4.

Convex hulls
Let X be a uniquely arcwise connected continuum. If S is a subset of X, we denote by [S] the intersection of all arcwise connected subsets containing S, and call it the convex hull of S in X. Clearly, [S] is the minimal one among all the arcwise connected subsets which contain S. We remark here that [S] need not be compact in general. If S = {a, b}, we also denote by [a, b] the convex hull of S, which is just the unique arc in X connecting a The following lemma is clear.
which is a tree.

Dendrites
Let X be a dendrite and let x ∈ X. We use ord(x, X) to denote the cardinality of the set of all components of X − {x}, which is called the order of x in X. The point x is a cut point if ord(x, X) ≥ 2; is a branch point if ord(x, X) ≥ 3; is an end point if ord(x, X) = 1. For a nondegenerate dendrite X, there are at most countably many branch points, there are uncountably many cut points, and there always exist end points. One may consult [7] for more properties about dendrites. The following two corollaries follow immediately from Proposition 2.3.
.., f n be homeomorphisms on a nondegenerate dendrite X, for some positive integer n. Suppose e is an endpoint of X such that f i (e) = e for all i = 1, ..., n. Then there are u, v = e ∈ X such that Corollary 2.5. Let f be a homeomorphism on a nondegenerate dendrite X. Suppose e is an endpoint of X such that f (e) = e. Then there is a sequence Let X and Y be metric spaces and let f :  . If G is a countable amenable group acting on a dendrite X, then either G has a fixed point or has a 2-periodic point in X.

Rays and lines
and ∩ ∞ n=0 ψ([n, +∞)) contain at least two points; is called one-sided-oscillatory if it is oscillatory but not bi-sided-oscillatory; is called nonoscillatory if it is not oscillatory.
We should note that if φ 1 and φ 2 are two rays with φ 1 ([0, +∞)) = φ 2 ([0, +∞)), then φ 1 and φ 2 have the same types of oscillation. The same conclusion is true for lines. One may consult [3] for more information about rays (the notion "ray" is called "quasi-arc" in [3]). Definition 2.9. Let X be a compact metric space. Let R be a ray in X and let L be a line in X. We say that L is an extension of R if there is a continuous injection φ : (−∞, +∞) → X such that L = φ((−∞, +∞)) and R = φ([0, +∞)).
The following lemma is clear.
Lemma 2.10. Let X be a uniquely arcwise connected continuum. Let R be a ray in X and let φ : [0, +∞) → X be a continuous injection such that (a, b), then R can be extended to a line L in X.

Quasi-retractions
Let X be a uniquely arcwise connected continuum. Let Y be either a tree, or a oscillatory ray, or a bi-sided oscillatory line contained in X. Then, by the uniquely arcwise connectivity, for every x ∈ X, there is a unique y ∈ Y such that [x, y] ∩ Y = {y}; we denote y = r Y (x), and call the map r Y : X → Y, x → r Y (x) the quasi-retraction from X onto Y . We should note that r Y is not continuous in general.
Lemma 2.12. Let X be a uniquely arcwise connected continuum. Let Y be either a tree, or a oscillatory ray, or a bi-sided oscillatory line contained in X. If Z is an arcwise connected subset of Y , then r −1 Y (Z) is an arcwise connected Borel measurable subset of X.
and Z is arcwise connected, we know r −1 Y (Z) is arcwise connected. For the measurability of r −1 Y (Z), one may consult [6].
3 Induced actions on dendrites

Convex metrics and their completions
Let X be a uniquely arcwise connected space (need not be compact).
Then T is an arcwise connected subset of X. Clearly, T is also the union of infinitely many arcs I i (i = 1, 2, 3, ...) with I i ∩ I j being a point or empty for any i = j. Without loss of generality, we may suppose that It is direct to check that d is a convex metric on T . Let T be the completion of T with respect to the metric d. We still use d to denote the naturally induced metric on T . Proof. Claim A. (T , d) is compact. Indeed, for every ǫ > 0, there is some n such that Since T n is a tree, there is a finite set {x 1 , ..., x m } ⊂ T n for some positive integer m, such that is the open ball with center x i and radius ǫ 2 under the metric d. For x ∈ T − T n , let y = r Tn (x). By (3.3), there is some x i with y ∈ B d (x i , ǫ 2 ). By (3.2) and the definition of d, we have . So T is totally bounded, which implies T is compact.
Claim B. (T , d) is locally connected. Indeed, for every ǫ > 0 there is some n such that Noting that T n is a tree, there are finitely many arcs C 1 , ..., C m such that , for each i = 1, ..., m.
Then D i is arcwise connected by Lemma 2.12. From (3.4) and (3.5), for all x, y ∈ D i , we have It follows from [7, 8.4] that T is locally connected.
Claim C. (T , d) is tree-like. Indeed, for every ǫ > 0, there is n such that Noting that for every ǫ ′ > 0, by the convexity of d, we always have d(r Tn (x), r Tn (y)) ≤ d(x, y) < ǫ ′ , whenever x, y ∈ T with d(x, y) < ǫ ′ . This shows that r Tn : (T, d) → (T n , d) is uniformly continuous. So, r Tn can be extended to a continuous map We want to show that r ǫ is an ǫ-map. Otherwise, there are x, y ∈ T with d(x, y) > ǫ and r ǫ (x) = r ǫ (y). Then by the continuity of r ǫ and the density of T in T , there are x ′ , y ′ ∈ T such that d(x, x ′ ) < ǫ 5 , d(y, y ′ ) < ǫ 5 , and d(r ǫ (x ′ ), r ǫ (y ′ )) < ǫ 5 . So, by (3.6), we have which is a contradiction. By the arbitrariness of ǫ, we get that (T , d) is tree-like. It follows from Claim A, Claim B, Claim C, and Theorem 2.6 that (T , d) is a dendrite.
Remark 3.2. The topology induced by d on T may not be the subspace topology of T induced from X; that is, the inclusion i : (T, d) → X, x → x, may not be an embedding in general.
T n , then the completion of T with respect to the metric d defined above is homeomorphic to the graph " H ".

Induced actions
Let X be a uniquely arcwise connected continuum. Let G be a countable amenable group acting on X. Suppose G = {g i : i = 1, 2, 3, ...}. Take a point p ∈ X. For each positive integer n, let S n = {g i (p) : i = 1, ..., n} and let T n = [S n ]. Then we get an increasing sequence of trees: . Then T is a G-invariant uniquely arcwise connected subset of X. We assume that T is not a tree. Then by deleting some T i 's in (3.8) and renumbering the remaining T i 's, we can assume that the sequence in (3.8) is strictly increasing. It follows from Proposition 3.1 that the completion (T , d) of T with respect to the metric d defined in (3.1) is a dendrite.
Proposition 3.4. The action of G on (T, d) is uniformly continuous with respect to the metric d.
Proof. Let g ∈ G. For every ǫ > 0, there is m such that Take a sufficiently large n so that S n ⊃ S m ∪ g −1 S m . Then g(S n ) ∩ S n ⊃ S m and (3.10) g(T n ) ∩ T n ⊃ T m .
From Proposition 3.4, we know that every g ∈ G can be extended to a continuous map g : (T , d) → (T , d). Proof. We need only to show that g is injective. By the definition of T , for any x = y ∈ T , there are Then d(g(x), g(y)) > d(g(x ′ ), g(y ′ )) > 0 by the convexity of d. This implies that g is injective.
From Proposition 3.3 and Proposition 3.5, we obtain an action of G on the dendrite (T , d) by homeomorphisms, which is called the induced action from the G-action on T .

Proof of the main theorem
In this section, we start to prove Theorem 1.2. Let X be a uniquely arcwise connected continuum and let G be a countable amenable group. We want to show that every G-action on X has either a fixed point or a 2-periodic point in X.
Altogether, we finish the proof of Theorem 4.1 under the assumption that G is finitely generated. Now, suppose that G is not finitely generated. For any finite subset F of G, let F be the subgroup of G, which is generated by F . Define Then X F is a nonempty closed subset of X. If F ′ is another finite subset of G, then X F ∩ X F ′ = X F ∪F ′ = ∅. Thus the family of compact sets {X F : F is a finite subset of G} has the finite intersection property. Hence