Free Topological G-Groups

Michael Megrelishvili* Department of Mathematics and Computer Science, Bar-Ilan University, 52900 Ramat-Gan, Israel

Necessary and sufficient conditions for the equivariant embeddability into: topological G-groups, G-linear spaces, homogeneous spaces, are obtained. In particular, free topological G-groups are investigated. A “regionally-proximal type” relation enables us to find compact homogeneous G-spaces G/H such that the free topological G-group over G/H is trivial. We construct also a G-group which is not G-linearizable into a locally convex linear G-space. Using J. West’s theorem about extension of actions from Z-subspaces to an action on the Hilbert cube, we prove that for every ℵ_{0}-bounded group G and a Gcompactificable space X there exists an equivariant embedding of ⟨G, X⟩ into ⟨H(I^{τ}), I^{τ} ⟩ where τ ≤ w(X) · w(G). In particular, Uspenskiĭ ’s theorem on the universality of H(I^{ℵ_{0}}) is improved.

1. Conventions and Known Facts.

In this paper, all topological spaces are Tychonoff (i.e., are completely regular and Hausdorff) and all linear spaces are real locally convex. A topological transformation group (a ttg) or for short: a G-space is a system ⟨G, X, α⟩ in which G is a topological group, X is a topological space and α: G × X → X is a continuous action. As usual, we write gx instead of α(g, 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. Denote by Comp^{G} and [Tych]^{G} the classes of all compact and all Tychonoff G-spaces respectively. A G-space X is called G-automorphic (or: a G-group) if X is a topological group and each α^{g} is an automorphism. In the case of a linear space X and linear α^{g} − s, we obtain the definition of a linear G-space [31].

Let H be a closed subgroup of G and G/H be the space of all left cosets endowed with the quotient topology. The triple ⟨G, G/H, α_{ℓ}⟩, where α_{ℓ}(g, sH) = gsH is called homogeneous.

Typeset by AMS-TEX

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

1991 Mathematics Subject Classification. 22A05, 54H15.

A ttg ⟨G, X, α⟩ will be called:

(a) G-compactificable, or G-Tychonoff if X is a G subspace of a compact G-space;

(b) G-homogenizable, if there exists an equivariant embedding of ⟨G, X, α⟩ into a homogeneous triple ⟨P, P/H, α_{ℓ}⟩ (i.e., there exists a topological group embedding φ: G ↪ P and a topological embedding f: X ↪ P/H such that α_{ℓ} |_{φ(G)×f(X)}= α).

(c) G-automorphizable, if X is a G-subspace of an automorphic G-space.

(d) G-linearizable, if X is a G-subspace of a linear G-space. Denote the corresponding classes by Tych^{G}, HTych^{G}, ATych^{G} and LTych^{G} respectively. Our aim will the comparision of these classes. Lemma 1.1. Let ⟨G, X, α⟩ be an automorphic triple, and let Xλ_{α}G denote the corresponding topological semidirect product. Then ⟨G, X, α⟩ is embedded into the homogeneous triple ⟨Xλ_{α}G, (Xλ_{α}G)/G, α_{ℓ}⟩, where G is identified with the subgroup {e} × G.

Proof. Consider the mapping q: X → (Xλ_{α}G)/G defined by the rule q(x) = xG. According to [25, Prop. 6.17(a)], q is a homeomorphism. Moreover, it can be easily checked that the restriction of α_{ℓ} on G × X: = G × q(X) is α. □

Actually the same trick was used (non-explicitly) by Ta-Sun Wu in [33, p.512]. Lemma 1.2. (J. de Vries [28]) Every homogeneous triple ⟨G, G/H, α_{ℓ}⟩ is G-Tychonoff.

Proposition 1.3. For every topological group G holds:

LTych^{G} ⊂ ATych^{G} ⊂ HTych^{G} ⊂ Tych^{G} ⊂ [Tych]^{G}.

Proof. Non-trivial inclusions follow from Lemmas 1.1 and 1.2. □

If G is locally compact (G ∈ LC), then every G-space is G-linearizable (see for example [26, 31]), and, hence all classes from Proposition 1.3 coincide. Example 1.4. [16]. In general, Tych^{G} ≠ [Tych]^{G}. There exists a continuous action α of a group G on J(ℵ_{0}) (the so-called hedgehog space of spininess ℵ_{0} [11]) such that ⟨G, J(ℵ_{0}), α⟩ is not G-Tychonoff. Note that in this example G and J(ℵ_{0}) are both Polish (i.e., are separable complete-metrizable).

Our main examples 3.7 and 4.7 will show that, in general, ATych^{G} ≠ HTych^{G} and LTych^{G} ≠ ATych^{G} .

Furthermore, 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) ≠ ∅.

The finest compatible uniformity we denote by μ_{max}.

Due to [25] the left, right, upper and lower uniformities on a topological group will be denote by L, R, L ∨ R, L ∧ R respectively. An action α on a uniform space (X, μ) is saturated if each g-transition is uniformly continuous.

A system ⟨G, (X, μ), α⟩ is called:

(a) bounded [29] (or: motion equicontinuous [8]) if for every ε ∈ μ there exists U ∈ N_{e}(G), satisfying (x, gx) ∈ ε for every (g, x) ∈ U × X.

(b) quasibounded [17] if for every ε ∈ μ there exists a pair (δ, U) ∈ μ × N_{e}(G) such that (gx, gy) ∈ ε whenever (x, y) ∈ δ and g ∈ U.

For example, if R(G/H) denotes the right uniformity on the coset space G/H, then the action α_{ℓ} is R(G/H)-bounded [28].

We will say that (X, μ) is a uniform G-space and write: ⟨G, (X, μ), α⟩ ∈ Unif^{G}, (or: (X, μ) ∈ Unif^{G}) if α is μ-quasibounded, μ-saturated and continuous. By [8] every compact G-space is bounded with respect to its unique uniformity. Therefore, Comp^{G} ⊂ Unif^{G} . Recall some useful facts concerning the class Unif^{G} . Fact 1.5. ([15, Th. 2.2], [18, Th. 1.5]). Let α: G × X → X be a continuous action. The following statements are equivalent:

(i) ⟨G, X, α⟩ ∈ Tych^{G}; (ii) ⟨G, (X, μ), α⟩ ∈ Unif^{G} for a certain compatible μ; (iii) ⟨G, (X, ξ), α⟩ is quasibounded for a certain compatible ξ.

Fact 1.6. ([17, Th. 1.2]). Let X be a topological group and α: G × X → X be a continuous action of a topological group G on X by automorphisms. Then ⟨G, (X, μ), α⟩ ∈ Unif^{G} for every μ ∈ {L, R, L ∨ R}.

Let G be a topological group and U ∈ N_{e}(G). Denote by Ũ the set of all pairs (x, y) ∈ G × G such that y ∈ UxU. Then {Ũ: U ∈ N_{e}(G)} is a base of the lower uniformity on G (cf. [25, Def. 2.5]). Using this description, it is easy to check that Fact 1.6 is true also for μ = L ∧ R.

For every topological group G there are several natural actions:

(a) ⟨G, G, α_{ℓ}⟩, α_{ℓ}(g, x) = gx;

(b) ⟨G, G, α_{r}⟩, α_{r}(g, x) = xg^{−1};

(c) ⟨G, G, α_{#}⟩, α_{#}(g, x) = gxg^{−1};

(d) ⟨G × G, G, π⟩, π((g_{1}, g_{2}), x) = g_{1}xg_{2}^{−1}. Elementary verification shows that ⟨G × G, (G, μ), π⟩ ∈ Unif^{G} for any μ ∈ {L, R, L ∨ R, L ∧ R}. This implies (by Fact 1.5) that for the actions α_{ℓ}, α_{r}, α_{#} there exists a common compact G-extension of G.

Denote by AUnif^{G} and LUnif^{G} the classes of all L ∨ R-uniform G-subspaces in automorphic and linear G-spaces respectively. Then, LUnif^{G} ⊂ AUnif^{G} ⊂ Unif^{G} . Indeed, AUnif^{G} ⊂ Unif^{G} directly follows from Fact 1.6 and the second inclusion is trivial. Fact 1.7. [17, Th. 1.10]. Suppose that G is Baire and α: G × X → X is a d-saturated action on a metrizable uniform space (X, d) such that for a certain dense subset Y ⊂ X orbit mapping α_{y}: G → X is continuous for every y ∈ Y. Then: (a) α is continuous;

(b) ⟨G, (X, d), α⟩ ∈ Unif^{G}; (c) X is G-Tychonoff.

Corollary 1.8. Let G be Baire and α: G×X → X be an action on a metrizable group by continuous automorphisms. Suppose that orbit mapping α_{y}: G → X is continuous for every y ∈ Y, where Y topologically generates X. Then α is continuous.

The last result essentially improves Proposition 2 from [6, VIII, §2.1].

Fact 1.9. [18, Th. 3.1]. Let ⟨G, (X, μ), α⟩ ∈ Unif^{G} . Denote by (X̂, μ̂) and Ĝ the completions of (X, μ) and (G, L∨R). Then there exists a continuous action α̂: Ĝ×X̂ → X̂ which extends α and satisfies ⟨Ĝ, (X̂, μ̂), α̂⟩ ∈ Unif^{G} .

2. West’s Theorem and Equivariant Homogenization.

For each compact space X, the group H(X) of all self-homeomorphisms, equipped with the compact-open topology, continuously acts on X by the rule π(h, x) = h(x). For a metrizable compact X, the group H(X) is Polish. If, in addition, π is transitive, then Effros’s Theorem [9] yields that the triple ⟨H(X), X, π⟩ is homogeneous.

Lemma 2.1. Let τ be an infinite cardinal and I^{τ} be the Tychonoff cube. Then the triple ⟨H(I^{τ}), I^{τ} , π⟩ is homogeneous.

Proof. The arguments preceding the Lemma and Theorem 4.1 from [5, p.104] show that ⟨H(I^{ℵ_{0}}), I^{ℵ_{0}}, π⟩ is homogeneous. By Exercise 3 of §5 in [25], the product ttg ⟨(H(I^{ℵ_{0}}))^{τ} , (I^{ℵ_{0}})^{τ} , π^{τ} ⟩, where π^{τ} is defined coordinate-wise, is also homogeneous. We can identify (I^{ℵ_{0}})^{τ} with I^{τ} and (H(I^{ℵ_{0}}))^{τ} with a topological subgroup of H(I^{τ}). Now the following claim completes the proof.

Claim. Let π: G × X → X be a continuous transitive action and P be a topological subgroup of G such that ⟨P, X, π |_{P ×X}⟩ is homogeneous. Then ⟨G, X, π⟩ is homogeneous too.

Proof of Claim. Let x ∈ X. By our assumption, the orbit mapping (π |_{P ×X})_{x} : P → X is open. For every nbd U ∈ N_{e}(G), the set (π |_{P ×X})_{x} (U) belongs to N_{x}(X) and is contained in π_{x}(U). Then π_{x}(U) ∈ N_{x}(X). Therefore, the action π is micro-transitive in the sense of [1]. This implies the openness of π_{x} (see, for example the proof (a) ⇒ (b) of Lemma 1 in [1]).

Remark 2.2. It should be mentioned that there exists (see [4]) a compact space X, such that the natural action π: H(X) × X → X is transitive but non-Effros. Due to our terminology, this means that ⟨H(X), X, α⟩ is not homogeneous. By the claim, the same is true for every topological subgroup G of H(X) transitively acting on X. Moreover, by Theorem 3.2 from [12], we can suppose that G carries an arbitrary group topology that leaves α |_{G×X} continuous.

Recall that a topological group G is called τ-bounded [14] (we write: ℓ^{u}(G) ≤ τ) if for every V ∈ N_{e}(G) there exists a subset S with the cardinality |S| ≤ τ such that SV = G. As is well-known, G is τ-bounded iff it is a topological subgroup in a product of groups with weight ≤ τ.

Fact 2.3. [17]. For every G-Tychonoff space X there exists a compact G-extension Y of weight w(Y) ≤ w(X) · ℓ^{u}(G).

A weaker version (for a locally compact G, and replacing ℓ^{u}(G) by ℓ(G) (= the Lindelöf degree of G) is obtained in [30].

Every continuous action α: G × X → X induces the continuous action G/ ker α × X → X, where ker α = {g ∈ G | gx = x for every x ∈ X}. We say that α is topologically exact (for short: t-exact) [19], if α is algebraically exact (i.e., ker α = {e}) and there is no strictly coarser group topology on G, that makes α continuous.

Lemma 2.4. Let ⟨G, X, α⟩ ∈ Tych^{G}. Then there exists a ttg ⟨G, Y, π⟩ ∈ Comp^{G} and a G-embedding f: X ↪ Y, such that w(Y) = w(X) · w(G) and π is t-exact.

Proof. Consider the action α_{ℓ}: G × G → G and the natural action of G on the disjoint sum X ⊔ G. Clearly, this action is t-exact. Since ⟨G, G, α_{ℓ}⟩ ∈ Tych^{G} (see [8] or Lemma 1.2). Then, X ⊔ G is G-Tychonoff as well. Finally, we apply Fact 2.3. □

J. West’s Theorem [32]. Any continuous action of a topological group G on a closed Z-subset of the Hilbert cube I^{ℵ_{0}} can be extended to a continuous action of G on I^{ℵ_{0}}.

Theorem 2.5. Let ⟨G, X, α⟩ ∈ Tych^{G} and w(G) · w(X) ≤ ℵ_{0}. Then ⟨G, X, α⟩ is embedded into ⟨H(I^{ℵ_{0}}), I^{ℵ_{0}}, π⟩.

Proof. By Lemma 2.4, without loss of generality, we may suppose that α is t-exact and X is a metrizable compact G-space. Embed X to I^{ℵ_{0}} as a Z-space, and use West’s Theorem. Then there exists a continuous action ⟨G, I^{ℵ_{0}}, α^{∗}⟩ such that algebraically G is a subgroup of H(I^{ℵ_{0}}) and α^{∗} |_{G×X}= α. Since α^{∗} is continuous, then the given topology σ on G contains the compact-open topology σ^{c} defining with respect to the action α^{∗}. The t-exactness of α (and hence of α^{∗}) implies σ = σ^{c}. □

Corollary 2.6. (Uspenskiǐ [27]). H(I^{ℵ_{0}}) is a universal separable metrizable topological group.

Theorem 2.7. Let G be an ℵ_{0}-bounded group. Then for every G-Tychonoff space X there exists an equivariant embedding of ⟨G, X⟩ into ⟨H(I^{τ}), I^{τ} ⟩ where τ ≤ w(X)·w(G).

Proof. It suffices to show that the given ttg ⟨G, X, α⟩ is embedded into ⟨(H(I^{ℵ_{0}}))^{τ} , (I^{ℵ_{0}})^{τ} ⟩ (see proof of Lemma 2.1). By Lemma 2.4 we can assume that α is t-exact and X is compact. Using the equivariant (generalization of Mardešic’s) approximation Theorem [17, Th. 2.19], we can represent X as a G-limit of an inverse G-system {⟨G, X_{i}, α_{i}⟩ | i ∈ M} (|M| ≤ τ), consisting of compact metrizable G-spaces X_{i}. Denote by f_{i} the i-th projection X → X_{i}, and by φ_{i} the canonical homomorphism G → G/ ker α_{i}. As in the proof of Theorem 2.5, the topology σ_{i} of G/ ker α_{i} contains the compact-open topology σ_{i}^{c}, defining with respect to the action α_{i}. Since w(X_{i}) ≤ ℵ_{0}, then w(G/ ker α_{i}, σ_{i}^{c}) ≤ ℵ_{0}. By Theorem 2.5, there exists an equivariant embedding

(ψ_{i}, h_{i}): ⟨(G/ ker α_{i}, σ_{i}^{c}), X_{i}⟩ ↪ ⟨H(I^{ℵ_{0}}), I^{ℵ_{0}}⟩ (i ∈ M).

Consider the diagonal products:

h̃ = ∏_{i∈M}(h_{i} ∘ f_{i}): X → (I^{ℵ_{0}})^{τ} , ψ̃ = ∏_{i∈M}(ψ_{i} ∘ φ_{i}): G → (H(I^{ℵ_{0}}))^{τ} .

Since X = lim_{←}{X_{i}: i ∈ M}, then h̃ is a topological embedding. Observe that the topological subgroup ψ̃(G) acts continuously on h̃(X). Since h̃ at the same time is equivariant, then the t-exactness of α implies that ψ̃ is a topological group embedding. □ Corollary 2.8. For every ℵ_{0}-bounded group G holds Tych^{G} = HTych^{G} .

Question 2.9. Is it true that Tych^{G} = HTych^{G} for every G?

Probably, the most natural candidate to be a counterexample is the above-mentioned (Remark 2.2) example from [4].

Concerning Theorem 2.7, let us note that a related result for an LC σ-compact group G (but with additional important properties) is proved in [2].

3. Free Topological G-Groups.

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.

Let ⟨G, X, α⟩ be a ttg. We will say that an automorphic 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 automorphic G-space P there exists an unique continuous G-homomorphism φ̃: F_{α}(X) → P such that φ̃ ∘ i_{α} = φ. If μ is an 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 (see for example, [13]), 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. Clearly, i_{α} is a topological (uniform) embedding iff ⟨G, X, α⟩ ∈ ATych^{G} (respectively : ⟨G, (X, μ), α⟩ ∈ AUnif^{G}). In this section we show that, in general, HTych^{G} ≠ ATych^{G} . Our main tool will be a “regionally proximal-type” relation. We start with the well known (at least for S = G) definition from topological dynamics.

Let ⟨G, X, α⟩ be a ttg, μ be a uniformity on X and S ⊂ G. A pair (a, b) ∈ X × X is called regionally S-proximal [7] 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 3.1. 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}. 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}. If S is symmetric and acts uniformly μ-equicontinuously on X (i.e., if the family {α^{g} | g ∈ S} is uniformly μ-equicontinuous), then Q_{S}^{p} = Q_{S} = Δ_{X}. Thus, if G ∈ LC and X is compact then Q_{∗}^{p} = Δ_{X}. In general, Q_{S} ≠ Q_{S}^{p} and Q_{∗}^{p} ≠ Q_{S}^{p}.

Example 3.2. 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. Therefore, Q_{G_{n}} is a proper subset of Q_{G_{n}}^{p} for each n ≥ 3.

Example 3.3. Define the homeomorphism 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 3.4. 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 3.5. 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 {a = x_{0}, x_{1}, …, x_{n} = b} satisfying Definition 3.1. 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 3.1, 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 automorphism, 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}. Eventually,

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}).

Clearly, 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 3.4 for the uniform G-mapping f = 1_{X}: (X, L ∨ R) → (X, R). □

Theorem 3.6. Let (X, μ) be a ∗-pseudoproximal G-space. Then every uniform Gmapping (X, μ) → (Y, ξ) into a G-group Y is trivial for each ξ ∈ {L, R, L ∨ R}. In particular, the free uniform G-group F_{α}(X, μ) is cyclic and discrete (that is, as trivial as possible).

Proof. Combine Lemma 3.4 and Theorem 3.5. □

Example 3.7. Let X = I^{n} be the n-dimensional cube, or let X = 𝕊^{n} be the ndimensional sphere (in both cases n ∈ ℕ). Then ⟨H(X), X, α⟩ is a regionally ∗- pseudoproximal ttg with respect to the unique uniformity on X. Then, by Theorem 3.5, the free topological G-group F_{α}(X) is cyclic and discrete. Since ⟨H(𝕊^{n}), 𝕊^{n}, α⟩ is a homogeneous triple, then we get that HTych^{G} ≠ ATych^{G} and Unif^{G} ≠ AUnif^{G} for G: = H(𝕊^{n}). This example for X = I answers the question posed by the author in [17, Problem 1.14].

Question 3.8. Under which conditions does the homogeneous triple ⟨G, G/H, α_{ℓ}⟩ belong to ATych^{G}?

This is so, for example, if H is a neutral [25] subgroup. Indeed, in such cases, Theorem 5.8 and Proposition 7.7 from [25] imply that the action α_{ℓ} of G on G/H is uniformly equicontinuous with respect to the quotient uniformity L/H. Therefore, by [22, Th.1.2] (or by our Proposition 3.11) G/H is even G-linearizable.

Question 3.9. Under which conditions does the free uniform G-group F_{α}(X, μ) coincide with the free uniform group F (X, μ) over X?

Fact 3.10 [17]. (For a stronger version see [18, Lemma 2.1]). Let an action α: G×X → X be quasibounded with respect to a uniformity μ on X and let each g-transition be continuous. Suppose that orbit mapping α_{y}: G → X is continuous for each y ∈ Y, where Y is dense in X. Then α is continuous.

Proposition 3.11. Let α: G×X → X be a saturated action on a uniform space (X, μ). Suppose that a certain V ∈ N_{e}(G) acts μ-uniformly equicontinuous. 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) [20, 23] follows that V acts L ∨ Runiformly equicontinuously on F (X, μ). In particular α̃ is L∨R-quasibounded. By Fact 3.10, α̃ is continuous. Obviously, this implies that F_{α}(X, μ) = F (X, μ). Essentially the same proof works for L(X, μ) using [24]. □

Pestov [21] proved the continuity of the associated action α̃: G × F_{ν}^{b}(X) → F_{ν}^{b}(X) for the uniformly equicontinuous 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 [22, Th. 1.2].

Lemma 3.12. If G ∈ LC and ⟨G, (X, μ), α⟩ ∈ Unif^{G}, then each compact subset of G acts μ-uniformly equicontinuous. In particular, this holds for μ = μ_{max}.

Proof. The standard compactness arguments prove the first assertion. In order to apply it to our second assertion, observe that ⟨G, (X, μ_{max}), α⟩ ∈ Unif^{G} (see [18, Pr. 3.7]). □

Proposition 3.13. If G ∈ LC and ⟨G, (X, μ), α⟩ ∈ Unif^{G}, then F_{α}(X, μ) = F (X, μ) and L_{α}(X, μ) = L(X, μ). In particular, F_{α}(X) = F (X) and L_{α}(X) = L(X).

Proof. Apply Lemma 3.12 and Proposition 3.11. □

Concerning the Proposition 3.13, it should be mentioned that the continuity of the lifted actions on A(X) and L(X) (under the assumption G ∈ LC) has been established earlier by Eisenberg [10] and de Vries [31] respectively. Example 3.14. We give an example where i_{α}: X → F_{α}(X) is a topological embedding and the group F_{α}(X) only algebraically coincides with F (X). Let ℚ be the topological group of all rational numbers. Consider the ttg ⟨ℚ, (ℚ, μ_{max}), α_{ℓ}⟩. Then μ_{max} is complete and saturated, but there is no continuous non-trivial action of ℝ on ℚ. By Fact 1.9, α_{ℓ} is not μ_{max}-quasibounded. Then Fact 1.6 implies that i_{α}: (ℚ, μ_{max}) → (F_{α_{ℓ}}(ℚ), L ∨ R) is not a uniform embedding. Thus, F_{α_{ℓ}}(ℚ) ≠ F (ℚ). On the other hand, by Proposition 3.11, F_{α_{ℓ}}(ℚ, L) = F (ℚ, L). Since F (ℚ, L) is, in particular, algebraically free over ℚ, then clearly F_{α_{ℓ}}(ℚ) is also algebraically free over ℚ.

Question 3.15. Is this true that if ⟨G, X, α⟩ ∈ ATych^{G}, then F_{α}(X) is algebraically free over X?

4. G-Linearizations. Definition 4.1. Let μ be the uniformity generated by a system P = {ρ_{i}}_{i∈M} of pseudometrics on a space X. We say that an action α: G × X → X is weakly P-Lipschitz if for every (i, g) ∈ M × G there exists (c, j, V) ∈ ℝ × M × N_{g}(G) such that ρ_{i}(gx, gy) ≤ cρ_{j}(x, y) whenever ρ_{i}(x, y) ≠ 0 and g ∈ V. If ρ_{j} can be choosed in such a way that the inequality holds for every (g, x, y) ∈ V × X × X, then we say that the action is P-Lipschitz. If α is continuous and P-Lipschitz (resp.: weakly P-Lipschitz) then we write ⟨G, (X, P), α⟩ ∈ Lip^{G} (respectively: ∈ Lip_{w}^{G}). If each ρ_{i} is a metric then these concepts coincide.

Theorem 4.2. If ⟨G, (X, μ), α⟩ ∈ LUnif^{G} then there exists a system P = {ρ_{i}}_{i∈M} of pseudometrics generating μ such that ⟨G, (X, P), α⟩ ∈ Lip_{w}^{G}.

Proof. Let (X, μ) be a uniform G-subspace of a locally convex linear G-space E. Consider a system {p_{i}}_{i∈M} of seminorms on E which generates the usual structure of E. We may assume that for every pair i, j ∈ M there exists k ∈ M such that p_{i} + p_{j} ≤ p_{k}. Fix g ∈ G and i ∈ M. From the continuity of the action G × E → E at (g, 0_{E}), where 0_{E} denotes the zero-element, it follows that for a certain δ > 0, V ∈ N_{g}(G) and j ∈ M, the inequality p_{j}(a) ≤ δ implies p_{i}(ga) ≤ 1 for each (g, a) ∈ V × E. We can suppose that p_{i} ≤ p_{j}. Then p_{i}(x) ≠ 0 implies that p_{j}(x) ≠ 0. Since p_{j} (\frac{δ}{p_{j}(x)}x) = δ, for every x ∈ X with the property p_{i}(x) ≠ 0, then p_{i} (g (\frac{δ}{p_{j}(x)}x)) ≤ 1. Therefore, p_{i}(gx) ≤ \frac{1}{δ}p_{j}(x) for every (g, x) ∈ V × E where p_{i}(x) ≠ 0. Now, it is obvious that the system of pseudometrics {ρ_{i}}_{i∈M}, where ρ_{i}(x, y) = p_{i}(x − y) for x, y ∈ X, is the desired one. □

Theorem 4.3. If ⟨G, (X, P), α⟩ ∈ Lip^{G} for a certain collection P generating μ then ⟨G, (X, μ), α⟩ ∈ LUnif^{G}.

Proof. Without restriction of generality we can assume that X has a fixed point. Apply the Arens-Eells [3] embedding (X, {ρ_{i}}_{i∈M}) ↪ (E, {ρ̃_{i}}_{i∈M}). The explicit description of ρ̃_{i} − s shows that the associated action α̃: G × E → E is {ρ̃_{i}}_{i∈M}-Lipschitz. Since each orbit mapping G → E is continuous, then by Fact 3.10, α̃ is continuous. □

We write ⟨G, X, α⟩ ∈ BTych^{G} if X is a G-subspace of a Banach G-space. Analogously, ⟨G, (X, μ), α⟩ ∈ BUnif^{G} means that (X, μ) is a uniform G-subspace in a Banach G-space. Theorem 4.4. For a metrizable uniformity μ, the following conditions are equivalent:

(i) ⟨G, (X, μ), α⟩ ∈ BUnif^{G}; (ii) ⟨G, (X, d), α⟩ ∈ Lip^{G} (or, equivalently: ∈ Lip_{w}^{G}) for a metric d; (iii) There exists a countable cover ∪{S_{n}}_{n∈ℕ} = G such that:

a) Each S_{n} acts μ-uniformly equicontinuous.

b) A certain S_{n_{0}} has an interior point.

Proof. (i) ⇒ (ii) in fact, is the same as in Theorem 4.2. For (ii) ⇒ (iii), consider S_{n} = {g ∈ G | d(gx, gy) ≤ nd(x, y) for every x, y ∈ X}.

For (iii) ⇒ (i) take a metric ρ ≤ 1 on X which generates μ, and define inductively A_{0} = {e} and A_{n+1} = A_{n} ·A_{n} ∪S_{n+1}. Then {A_{n}}_{n∈ℕ} also satisfies (iii) and, in addition, holds A_{n} · A_{m} ⊂ A_{n+m}. From this inclusion easily follows that if we define

d(x, y) = sup_{n∈ℕ} \{\frac{1}{2^{n}} sup_{g∈A_{n}}\{ρ(gx, gy)\}\}

then d(gx, gy) ≤ 2^{n}d(x, y) for each (g, x, y) ∈ A_{n} × X × X. Using the arguments from the proof of Theorem 4.3, by the Arens-Eells construction, there exists an isometric G-embedding of (X, d) into a normed G-space E. Since LUnif^{G} ⊂ Unif^{G}, then Fact 1.9 completes the proof. □

Theorem 4.5. Let G be an LC σ-compact group, and X be a metrizable G-space. Then ⟨G, X, α⟩ ∈ BTych^{G} . Moreover, the following statements are equivalent:

(i) ⟨G, (X, μ), α⟩ ∈ BUnif^{G}; (ii) Each g-transition is μ-uniformly continuous and μ is metrizable.

Proof. (i) ⇒ (ii) is trivial. For (ii) ⇒ (i) use consequently Fact 1.7(b), Lemma 3.12 and apply Theorem 4.4. In order to check ⟨G, (X, μ), α⟩ ∈ BTych^{G}, if suffices to show that there exists a metrizable saturated uniformity μ. A stronger result can be found in [18, Th. 3.8]. So our proof is completed. □

Theorem 4.6. Let X be a compact metrizable G-space and G be a closed topological subgroup of H(X). Then the following statements are equivalent:

(i) G is LC and σ-compact; (ii) ⟨G, X, α⟩ ∈ BTych^{G} .

Proof. (i) ⇒ (ii) directly follows from Theorem 4.5. For (ii) ⇒ (i) use the implication (i) ⇒ (iii) from Theorem 4.4 and apply the Ascolli-Arzela Theorem. □

Theorems 4.4 and 4.5 have been proved also in [15]. We close the paper with the following principal counterexample. Example 4.7. There exists a G-group X which is not G-linearizable. This means that, in general, ATych^{G} ≠ LTych^{G} .

Construction. Let X be the measure algebra of all equivalence classes of Lebesgue measurable subsets of the closed interval I. Under the operation Δ (the symmetric difference) and the norm ∥A∥ = meas (A), X is a separable complete-metrizable topological Abelian group homeomorphic to the Hilbert space ℓ_{2} (see Theorem 7.1 and Corollary 7.2 in [5, VI. 7]). Let G be the group of all such homeomorphisms h: I → I that h and h^{−1} are Lipschitz and denote by α: G × I → I the action α(h, x) = h(x). Since each α^{h} is a measurable function, then the associated action α^{∗}: G × X → X is well defined. Moreover, each transition (α^{∗})^{h}: X → X is ∥ · ∥-uniformly continuous. Endow G with the topology τ_{∥ · ∥} of ∥ · ∥-uniform convergence. Then α^{∗} is continuous. It turns out that the constructed G-group X is not G-linearizable.

Proof. In order to show that X is a desired G-group, observe that the mapping φ: I → X, φ(t) = [0, t] is an isometric G-embedding. Therefore, it suffices to check that ⟨G, I, α⟩ is not G-linearizable. We will prove in fact that I, as a G-space, is not even G-mapped non-trivially into a linear G-space. Assume the contrary. Then, due to Theorem 4.2, α is weakly P-Lipschitz for a certain family P = {ρ_{i}}_{i∈M} of continuous pseudometrics on I containing a non-constant ρ_{i_{0}}. There exists a sequence (x_{n})_{n∈ℕ} in I converging in the usual topology to a point x_{0} such that ρ_{i_{0}}(x_{n}, x_{0}) ≠ 0 for every n ∈ ℕ. Indeed, if this is not so, then for every x ∈ I one can choose δ_{x} > 0 such that ρ_{i_{0}}(y, x) = 0 for every y ∈ B(x, δ_{x}), where B(x, δ_{x}) = {y ∈ I | |y − x| < δ_{x}}. There exists a finite cover {B(x_{k}, δ_{x_{k}})}_{k=1}^{n} of I, such that B(x_{k}, δ_{x_{k}}) ∩ B(x_{k+1}, δ_{x_{k+1}}) ≠ ∅ for each k ∈ {1, 2, …, n − 1}. Since ρ_{i_{0}} is a pseudometric, then by the triangle axiom, we get ρ_{i_{0}}(x, y) = 0 for every x, y ∈ I. This contradiction proves the existence of the abovementioned (x_{n})_{n∈ℕ}. Without any loss of generality we may assume that this sequence is monotonic. For simplicity suppose that (x_{n})_{n∈ℕ} is increasing and hence each x_{n} lies in [0, x_{0}).

By Definition 4.1 there exists a triple (c, ρ_{j}, V) from ℝ × P × N_{e}(G) such that V = V^{−1} and ρ_{(}x, y) ≠ 0 implies that

ρ_{i_{0}}(h(x), h(y)) ≤ cρ_{j}(x, y) ∀ h ∈ V.  (*)

Since ρ_{j} is a continuous pseudometric, then ρ_{j}(x_{n}, x_{0}) tends to zero. Thus we can choose a subsequence (x_{n_{k}})_{k∈ℕ} such that x_{k} < x_{n_{k}} and

2cρ_{j}(x_{n_{k}}, x_{0}) ≤ ρ_{i_{0}}(x_{k}, x_{0}) ∀ k ∈ ℕ.  (**)

For each k ∈ ℕ there exists a homeomorphism h_{k}: I → I satisfying:

(1) h_{k}(x_{k}) = x_{n_{k}};

(2) h_{k}(t) = t for every t ∈ [0, x_{k−1}] ∪ [x_{0}, 1];

(3) h_{k} is linear on [x_{k−1}, x_{k}] and [x_{k}, x_{0}]. Clearly, each h_{k} belongs to G. Moreover, (2) implies that

meas (h_{k}(A)ΔA) ≤ |x_{k−1} − x_{0}| ∀ A ∈ X.

Therefore, (h_{k})_{k∈ℕ} converges to the identity in G. Thus, h_{m} ∈ V for a certain m ∈ ℕ. Then, by (**), (*), (1), (2), we get

ρ_{i_{0}}(x_{m}, x_{0}) ≥ 2cρ_{j}(x_{n_{m}}, x_{0}) ≥ 2ρ_{i_{0}} (h_{m}^{−1}(x_{n_{m}}), h_{m}^{−1}(x_{0})) = 2ρ_{i_{0}}(x_{m}, x_{0}).

Hence, ρ_{i_{0}}(x_{m}, x_{0}) ≥ 2ρ_{i_{0}}(x_{m}, x_{0}) > 0. This contradiction completes the proof. □

Acknowledgemement. I would like to express my gratitude to Yu.M. Smirnov, J. de

Vries and V.G. Pestov for valuable remarks. I am also grateful to the referee for helpful

comments.

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