(To appear in: Annals of the New York Academy of Sciences, volume 788 (1996), pp. 64-70, PAPERS ON GENERAL TOPOLOGY AND APPLICATIONS (TENTH SUMMER CONFERENCE AT AMSTERDAM))

Free topological groups over (semi)group actions

M.G. Megrelishvili (Levy)* Department of Mathematics and Computer Science Bar-Ilan University, 52 900 Ramat-Gan, ISRAEL

Abstract. We study equivariant embeddability into G-groups. A new regionally proximal type relation introduced in the paper gives a necessary condition providing some counterexamples. We establish also some sufficient conditions (for locally compact acting semigroups G, for instance) improving results of M.Eisenberg and J. de Vries.

Introduction.

Our aim is to investigate the following question.

Question. Let a topological (semi)group G act continuously on a space X. When can X be equivariantly embedded (or at least, G-mapped non-trivially) into a topological group P in such a way that G continuously acts on P by endomorphisms (hence, by automorphisms if G is a group)?

For the particular case when P is a linear G-space see de Vries [11] and the references there.

The question leads us to the definition of the free topological G-group over a (semi)group action (see Definition 1). Our main result is Theorem 6 which enubles us to find compact coset G-spaces G/H such that the free topological G-group over G/H is trivial. Roughly speaking, this means that every continuous G-map of G/H into a G-group P would be “collapsed” into a point. This happens, for example, when G/H = 𝕊^{n} is the n-dimensional sphere where G is the group of all autohomeomorphisms of 𝕊^{n} endowed with the compact open topology. The main tool will be a new “regionally proximal type” relation (Definition 2) which generalizes the classical notion from topological dynamics.

Eisenberg [3] has shown that if a locally compact group G acts continuously on a Tychonoff space X then the induced “lifted” action G × A(X) → A(X) (which is

Typeset by AMS-TEX

*Partially supported by the Israel Ministry of Sciences, Grant No. 3505.

1991 Mathematics Subject Classification. 54H15,54H20, 22A05.

Key words and phrases. topological transformation semigroup, free topological group, locally compact semigroup, regionally proximal relation.

separately continuous for arbitrary G) on the free Abelian topological group A(X) is jointly continuous. A similar result, if A(X) is replaced by the free locally convex space L(X) is also true. This was remarked without proof in [3]. De Vries [11] proved it by categorical methods. We establish that analogous results remain true for the free topological group F (X) considering an arbitrary locally compact topological semigroup G.

Conventions.

Recall that a G-space or alternatively, a topological transformation (semi)group (abbreviated: tts, ttg) is a system ⟨G, X, α⟩ in which G is a topologized (semi)group, X is a topological space and α : G × X → X, α(g, x) = gx is a continuous action. As usual, this means that (gh)x = g(hx) for every g, h ∈ G and every x ∈ X. If G has an identity e (i.e., if G is a monoid) then we require ex = x for every x ∈ X. A g-transition is the mapping α^{g} : X → X, α^{g}(x) = gx and an x-orbit mapping is the mapping α_{x} : G → X, α_{x}(g) = gx. A G-space X will be called a G-group orG-endomorphic if X is a topological group and each α^{g} is a group endomorphism. If G is a group we use the term: G-automorphic. In the case of a linear space X and linear α^{g} − s, we obtain the known definition of a linear G-space [11].

The filter of all neighborhoods (nbd’s) at a point x in a space X is denoted by N_{x}(X). If μ is a compatible uniformity on a topological space X, then for every ε ∈ μ and A ⊂ X denote by ε(A) the set {y ∈ X | (x, y) ∈ ε, x ∈ A}. Subsets A, B will be called ε-near if ε(A) ∩ ε(B) ≠ ∅.

We denote the greatest compatible uniformity by μ_{max}.

Due to [10], the left, right and upper uniformities on a topological group will be denoted by L, R, L ∨ R respectively.

We say that an action α : G × X → X is locally uniformly equicontinuous if for every g ∈ G there exists V ∈ N_{g}(G) such that {α^{g}}_{g∈V} is uniformly equicontinuous.

Main results.

As usual, for a topological space X denote by F (X), A(X), L(X) the free topological group, the free Abelian topological group and the free locally convex space respectively.

Definition 1. Let ⟨G, X, α⟩ be a tts. We will say that an endomorphic triple ⟨G, F_{α}(X), α̃⟩ with a continuous G-mapping i_{α}: X → F_{α}(X) is the free topological G-group over X, if for every continuous G-mapping φ: X → P to an endomorphic G-space P there exists a unique continuous G-homomorphism φ̃: F_{α}(X) → P such that φ̃ ∘ i_{α} = φ. If μ is a uniformity on X, then considering uniform G-mappings and the upper uniformities on topological groups, we obtain the definition of the uniform free topological G-group over (X, μ). The corresponding universal morphism is denoted by i_{α}: (X, μ) → F_{α}(X, μ).

The (uniform) free locally convex G-space L_{α}(X) (respectively : L_{α}(X, μ))can be analogously defined.

An obvious equivariant generalization of the standard product procedure shows that the just defined free G-objects always exist. However, it turns out that the embedding problem for i_{α} is much more complicated. We start with the well known definition from topological dynamics.

Let ⟨G, X, α⟩ be a tts, μ be a uniformity on X and S ⊂ G. A pair (a, b) ∈ X × X is called regionally S-proximal [1] and is indicated: (a, b) ∈ Q_{S}, if for every ε ∈ μ and arbitrary nbd’s O_{1} ∈ N_{a}(X), O_{2} ∈ N_{b}(X) there exists g ∈ S such that gO_{1} and gO_{2} are ε-near. Otherwise, (a, b) is said to be regionally S-distal. X is called regionally S-distal if Q_{S} = Δ_{X}: = {(x, x) | x ∈ X}. The following definition seems to be new.

Definition 2. (i) We say that a pair (a, b) ∈ X × X is regionally S-pseudoproximal and write : (a, b) ∈ Q_{S}^{p} (or: (a, b) ∈ Q_{S}^{p}(X, μ)) if there exists a finite set {a = x_{0}, x_{1}, …, x_{n} = b} with the following property:

(∗^{S}) for every ε ∈ μ and arbitrary nbd’s O_{i} ∈ N_{x_{i}}(X), i ∈ {0, 1, …, n} there exists

g ∈ S such that gO_{i} and gO_{i+1} are ε-near, for every i ∈ {0, 1, …, n − 1}. (ii) Let G be a monoid.A pair (a, b) will be called regionally ∗-pseudoproximal if

(a, b) ∈ Q_{V}^{p} for every V ∈ N_{e}(G). This defines a relation Q_{∗}^{p} = ∩{Q_{V}^{p} | V ∈ N_{e}(G)}. If Q_{∗}^{p} = X × X or Q_{∗}^{p} = Δ_{X}, then we say that X is regionally∗-pseudoproximal, or regionally ∗-pseudodistal respectively.

Obviously, Q_{S}^{p} and Q_{∗}^{p} are reflexive symmetric relations on X and always Q_{S} ⊂ Q_{S}^{p}, Q_{∗}^{p} ⊂ Q_{S}^{p}. In general, Q_{S} ≠ Q_{S}^{p}, and Q_{∗}^{p} ≠ Q_{S}^{p}.

Example 3. Let G_{n} = \{h ∈ H(I) | h(x_{i}) = x_{i}, x_{i} = \frac{i}{n}, i ∈ \{0, 1, …, n\}\} be the topological subgroup of H(I). Consider the ttg ⟨G_{n}, I, α⟩ and the canonical uniformity on I. Then, for every natural n ≥ 3, the elements 0 and 1 are regionally G_{n}-distal. On the other hand, every pair (a, b) ∈ I × I is regionally ∗-pseudoproximal. In particular, Q_{G_{n}} is a proper subset of Q_{G_{n}}^{p} for each n ≥ 3.

Example 4. Define the homeomomorphism h : I → I by the rule

h(x) = \{ \begin{matrix}3x^{2}, & 0 ≤ x ≤ \frac{1}{3} \\ \frac{1}{3} + \frac{1}{3} \sqrt{3x − 1}, & \frac{1}{3} ≤ x ≤ \frac{2}{3} \\ 3x^{2} − 4x + 2, & \frac{2}{3} ≤ x ≤ 1. \end{matrix}

Consider the cyclic group G = {h^{n}}_{n∈ℤ} and the natural action G × I → I. Since 0, \frac{1}{3}, \frac{2}{3}, 1 are fixed then clearly (0, 1) ∉ Q_{G}. On the other hand, elementary computations show that Q_{G}^{p} = I × I. Note also that Q_{∗}^{p} = Δ_{X} if G is discrete.

Lemma 5. If f: (X_{1}, μ_{1}) → (X_{2}, μ_{2}) is a uniform G-mapping, then (f×f)(Q_{S}^{p}(X_{1}, μ_{1})) ⊂ Q_{S}^{p}(X_{2}, μ_{2}) and (f × f)(Q_{∗}^{p}(X_{1}, μ_{1})) ⊂ Q_{∗}^{p}(X_{2}, μ_{2}).In particular, if (X, μ) is regionally ∗-pseudodistal, then every uniform G-subspace (Y, μ|_{Y}) is regionally ∗-pseudodistal.

Theorem 6. Let G be a topologized monoid.Then every G-group ⟨G, (X, μ), α⟩ is regionally∗-pseudodistal for each μ ∈ {L, R, L ∨ R}.

Proof. First we consider the case μ = R. Assuming the contrary, take a pair (a, b) ∈ Q_{∗}^{p} of distinct elements. Since X is a Hausdorff topological group and α is continuous, then we can choose nbd’s V_{0} ∈ N_{e}(X), U ∈ N_{e}(G) such that

V_{0} ∩ g(V_{0}ab^{−1}) = ∅  ∀ g ∈ U  (1)

Since Q_{∗}^{p} ⊂ Q_{U}^{p}, then (a, b) ∈ Q_{U}^{p}. Consider a finite set {x_{0}, x_{1}, …, x_{n}} satisfying Definition 2. Choose symmetric nbd’s V_{1}, V_{2} ∈ N_{e}(X) with the properties:

x_{0}x_{i}^{−1}V_{2}^{2} ⊂ V_{1}x_{0}x_{i}^{−1}  ∀ i ∈ {0, 1, …, n}  (2)
V_{1}^{n+1} ⊂ V_{0}  (3)

Due to Definition 2(i), we pick for ε: = {(x, y) ∈ X ×X | xy^{−1} ∈ V_{2}} an element g ∈ U such that g(V_{2}x_{i}) and g(V_{2}x_{i+1}) are V_{2}-near with respect to the right uniformity R on X. More precisely, there exist finite sequences {p_{0}, p_{1}, …, p_{n−1}}, {q_{1}, q_{2}, …, q_{n}} in V_{2} such that g(p_{i}x_{i})(g(q_{i+1}x_{i+1}))^{−1} ∈ V_{2} for every i ∈ {0, 1, …, n − 1}.

Since α^{g} is an endomorphism, then

g(p_{i}x_{i}x_{i+1}^{−1}q_{i+1}^{−1}) ∈ V_{2}  ∀ i ∈ {0, 1, …, n − 1}  (4)

Consider the element

z = g(p_{0}x_{0}x_{1}^{−1}q_{1}^{−1}) g(p_{1}x_{1}x_{2}^{−1}q_{2}^{−1}) ⋯ g(p_{n−1}x_{n−1}x_{n}^{−1}q_{n}^{−1}).

Since V_{2} ⊂ V_{1} by (2), then (4) and (3) imply z ∈ V_{2}^{n} ⊂ V_{1}^{n} ⊂ V_{0}. Clearly,

z = g(p_{0}x_{0}x_{1}^{−1}(q_{1}^{−1}p_{1})x_{1}x_{2}^{−1}(q_{2}^{−1}p_{2}) ⋯ (q_{n−1}^{−1}p_{n−1})x_{n−1}x_{n}^{−1}q_{n}^{−1})

and, q_{i}^{−1}p_{i} ∈ V_{2}^{−1}V_{2} = V_{2}^{2} for each i ∈ {1, …, n − 1}. Using (2) and the trivial cancellations of the form x_{0}x_{i}^{−1}x_{i}x_{i+1}^{−1} = x_{0}x_{i+1}^{−1}, (1 ≤ i ≤ n − 1) after n − 1 steps we get

z ∈ g(p_{0}V_{1}^{n−1}x_{0}x_{n}^{−1}q_{n}^{−1}) ⊂ g(V_{1}^{n}x_{0}x_{n}^{−1}q_{n}^{−1}).

Using (2) (for i = n), we obtain

z ∈ g(V_{1}^{n+1}x_{0}x_{n}^{−1}) = g(V_{1}^{n+1}ab^{−1}) ⊂ g(V_{0}ab^{−1}).

Thus, z ∈ V_{0} ∩ g(V_{0}ab^{−1}), which contradicts (1). This proves the case μ = R.

For μ = L, use the G-unimorphism (X, L) → (X, R), x → x^{−1} and if μ = L ∨ R, use Lemma 5 for the uniform G-mapping f = 1_{X}: (X, L ∨ R) → (X, R). □

Theorem 7. Let G be a topologized monoid and let (X, μ) be a ∗-pseudoproximal Gspace. Then every uniform G-mapping (X, μ) → (Y, ξ) into a G-group Y is constant for each ξ ∈ {L, R, L ∨ R}.In particular, the free uniform G-group F_{α}(X, μ) is cyclic and discrete.

Proof. Combine Lemma 5 and Theorem 6. □

Example 8. Let X = I^{n} be the n-dimensional cube, or let X = 𝕊^{n} be the n-dimensional sphere (in both cases n ∈ ℕ). Denote by H(X) the group of all autohomeomorphisms of X endowed with the compact open topology. Then ⟨H(X), X, α⟩ is a regionally ∗-pseudoproximal ttg with respect to the unique uniformity on X. Then, by Theorem 7, the free topological G-group F_{α}(X) is cyclic and discrete. It is remarkable that by Effros’s Theorem [2], 𝕊^{n} is a coset space of H(𝕊^{n}). If X = I then the example answers the question posed by the author in [4, Problem 1.14].

Question 9. Under which conditions is the G-space G/H automorphizable (= Gsubspace of an automorphic G-space) ?

This is so, for example, if H is a neutral [10] subgroup. Indeed, in such cases, Theorem 5.8 and Proposition 7.7 from [10] imply that the action α_{ℓ} of G on G/H is uniformly equicontinuous with respect to the quotient uniformity L/H. Therefore, by [7, Th.1.2] (or by our Proposition 12) G/H is even G-linearizable.

Question 10. Under which conditions does the free uniform G-group F_{α}(X, μ) coincide with the free uniform group F (X, μ) over X?

Lemma 11. (For a stronger version for groups, see [5, Lemma 2.1]). Let an action α: G × X → X be locally uniformly equicontinuous with respect to a uniformity μ on X and let orbit mapping α_{y}: G → X be continuous for each y ∈ Y, where Y is dense in X. Then α is continuous.

Proposition 12. Let α: G × X → X be a continuous and locally uniformly equicontinuous action on a uniform space (X, μ). Then F_{α}(X, μ) = F (X, μ) and L_{α}(X, μ) = L(X, μ).

Proof. Let α̃: G × F (X, μ) → F (X, μ) be the lifted action. Clearly, each g-transition α̃^{g} is continuous. Since X algebraically generates F (X, μ), then the continuity of orbit mappings α_{x}: G → X and of group operations in F (X, μ) imply that for each w ∈ F (X, μ) the orbit mapping α̃_{w}: G → F (X, μ) is continuous. From the constructive description of a neighborhood system of the identity in F (X, μ) [8] follows that V acts L ∨ R-uniformly equicontinuously on F (X, μ),provided that V acts uniformly equicontinuous on (X, μ). By Lemma 11, α̃ is continuous.Obviously, this implies that F_{α}(X, μ) = F (X, μ). Essentially the same proof works for L(X, μ) using [9]. □

Pestov [6] proved the continuity of the associated action α̃: G × F_{ν}^{b}(X) → F_{ν}^{b}(X) for the uniformly equicontinuous group action α: G × X → X, where F_{ν}^{b}(X) denotes the free uniform balanced (i.e., L = R) group in a variety ν. For an analogous “lifting” Theorem for a modification of the free locally convex spaces, see [7, Th. 1.2].

Lemma 13. Every continuous action α of a locally compact topological semigroup G on a Tychonoff space X is locally μ_{max}-uniformly equicontinuous.

Proof. Let a system S = {d_{k}}_{k∈K} of pseudometrics generate μ_{max} and let 𝔹 be the system of all compact subsets in G. Consider the family S^{𝔹} = {d_{k}^{C} | k ∈ K, C ∈ 𝔹} where

d_{k}^{C}(x, y) = sup{d_{k}(gx, gy) | g ∈ C}.

The compactness of C and the continuity of α imply that the system {α^{g} | g ∈ C} is d_{k}-equicontinuous for every k ∈ K. Then it is easy to see that uniformity θ generated by the system S^{𝔹} ∪ S is compatible with the original topology. If A, B ∈ 𝔹 then A · B ∈ 𝔹. Eventually, the given action is locally θ-uniformly equicontinuous. Finally, observe that the maximality of μ_{max} and the inclusion μ_{max} ⊂ θ imply μ_{max} = θ. □

Theorem 14. For every continuous action α of a locally compact topological semigroup G on a Tychonoff space X holds F_{α}(X) = F (X), A_{α}(X) = A(X), L_{α}(X) = L(X).

Proof. It is well known that F (X, μ_{max}) = F (X), A(X, μ_{max}) = A(X) and L(X, μ_{max}) = L(X). So we can apply Proposition 12 and Lemma 13. □

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E-mail: megereli@macs.biu.ac.il