Compact Metrizable Group Research Articles

Approximation properties in Lipschitz‐free spaces over groups

We study Lipschitz-free spaces over compact and uniformly discrete metric spaces enjoying certain high regularity properties — having group structure with left-invariant metric. Using methods of harmonic analysis we show that, given a compact metrizable group G $G$ equipped with an arbitrary compatible left-invariant metric d $d$ , the Lipschitz-free space over G $G$ , F ( G , d ) $\mathcal {F}(G,d)$ , satisfies the metric approximation property. We show also that, given a finitely generated group G $G$ , with its word metric d $d$ , from a class of groups admitting a certain special type of combing, which includes all hyperbolic groups and Artin groups of large type, F ( G , d ) $\mathcal {F}(G,d)$ has a Schauder basis. Examples and applications are discussed. In particular, for any net N $N$ in a real hyperbolic n $n$ -space H n $\mathbb {H}^n$ , F ( N ) $\mathcal {F}(N)$ has a Schauder basis.

A note on the Kawada–Itô theorem

A probability measure μ on a locally compact group G is said to be adapted if the support of μ generates a dense subgroup of G. A classical Kawada–Itô theorem asserts that if μ is an adapted measure on a compact metrizable group G, then the sequence of probability measures 1n∑k=0n−1μkn=1∞ weak∗ converges to the Haar measure on G. In this note, we present a new proof of Kawada–Itô theorem. Also, we show that metrizability condition in the Kawada–Itô theorem can be removed. Some applications are also given.

Equidistribution of random walks on compact groups

Let X1,X2,… be independent, identically distributed random variables taking values from a compact metrizable group G. We prove that the random walk Sk=X1X2⋯Xk, k=1,2,… equidistributes in any given Borel subset of G with probability 1 if and only if X1 is not supported on any proper closed subgroup of G, and Sk has an absolutely continuous component for some k≥1. More generally, the sum ∑k=1Nf(Sk), where f:G→R is Borel measurable, is shown to satisfy the strong law of large numbers and the law of the iterated logarithm. We also prove the central limit theorem with remainder term for the same sum, and construct an almost sure approximation of the process ∑k≤tf(Sk) by a Wiener process provided Sk converges to the Haar measure in the total variation metric.

Expansive actions of automorphisms of locally compact groups $${\varvec{G}}$$ on $$\hbox {Sub}_{{\varvec{G}}}$$

For a locally compact metrizable group $G$, we consider the action of ${\rm Aut}(G)$ on ${\rm Sub}_G$, the space of all closed subgroups of $G$ endowed with the Chabauty topology. We study the structure of groups $G$ admitting automorphisms $T$ which act expansively on ${\rm Sub}_G$. We show that such a group $G$ is necessarily totally disconnected, $T$ is expansive and that the contraction groups of $T$ and $T^{-1}$ are closed and their product is open in $G$; moreover, if $G$ is compact, then $G$ is finite. We also obtain the structure of the contraction group of such $T$. For the class of groups $G$ which are finite direct products of $\mathbb{Q}_p$ for distinct primes $p$, we show that $T\in{\rm Aut}(G)$ acts expansively on ${\rm Sub}_G$ if and only if $T$ is expansive. However, any higher dimensional $p$-adic vector space $\mathbb{Q}_{p^n}$, ($n\geq 2$), does not admit any automorphism which acts expansively on ${\rm Sub}_G$.

On convergent sequences in dual groups

We provide some characterizations of precompact abelian groups $G$ whose dual group $G_p^\wedge$ endowed with the pointwise convergence topology on elements of $G$ contains a nontrivial convergent sequence. In the special case of precompact abelian \emph{torsion} groups $G$, we characterize the existence of a nontrivial convergent sequence in $G_p^\wedge$ by the following property of $G$: \emph{No infinite quotient group of $G$ is countable.} Finally, we present an example of a dense subgroup $G$ of the compact metrizable group $\mathbb{Z}(2)^\omega$ such that $G$ is of the first category in itself, has measure zero, but the dual group $G_p^\wedge$ does not contain infinite compact subsets. This complements Theorem 1.6 in [J.E.~Hart and K.~Kunen, Limits in function spaces and compact groups, \textit{Topol. Appl.} \textbf{151} (2005), 157--168]. As a consequence, we obtain an example of a precompact reflexive abelian group which is of the first Baire category.

Some applications of conditional expectations to convergence for the quantum Gromov–Hausdorff propinquity

We prove that all the compact metric spaces are in the closure of the class of full matrix algebras for the quantum Gromov–Hausdorff propinquity. We also show that given an action of a compact metrizable group $G$ on a quasi-Leibniz quantum compact metric

Universal free action for compact groups

For every compact metrizable group G there is a free G-action on the Hilbert space which is universal for free G-actions on compact metric spaces.

Surjunctivity and topological rigidity of algebraic dynamical systems

Let$X$be a compact metrizable group and let$\unicode[STIX]{x1D6E4}$be a countable group acting on$X$by continuous group automorphisms. We give sufficient conditions under which the dynamical system$(X,\unicode[STIX]{x1D6E4})$is surjunctive, i.e. every injective continuous map$\unicode[STIX]{x1D70F}:X\rightarrow X$commuting with the action of$\unicode[STIX]{x1D6E4}$is surjective.

Mean ergodic theorem for amenable discrete quantum groups and a Wiener-type theorem for compact metrizable groups

We prove a mean ergodic theorem for amenable discrete quantum groups. As an application, we prove a Wiener type theorem for continuous measures on compact metrizable groups.

G-fibrant extensions and twisted products

Let G be a compact metrizable group and H its closed subgroup. We prove that every compact metrizable G-space over G/H admits a G-fibrant extension over G/H. This implies that the functor of twisted product G×H− takes H-fibrant extensions to G-fibrant extensions. In particular, the twisted product G×HS is a G-fibrant space when S is a compact H-fibrant space.

G-fibrations and twisted products

We prove that the functor of twisted product G×H− takes H-fibrations to G-fibrations when G is a compact metrizable (not necessarily Lie) group and H is its closed subgroup. This result is applied to the study of strong G-fibrations. In particular, we show that every G-map E→G/H is a strong G-fibration provided that E is a G-fibrant space.

Bayesian and Maximum Likelihood Solutions: An Asymptotic Comparison Related to Cost Function

Problem statement: Wald showed that the minimax solution is the Bayes ian solution with respect to the law a priori the worst. We try to establish a similar result by comparing the Bayesian solution and the solution of maximum likel ihood when the parameter space is a compact metrizable group. Approach: we take as a priori law Haar measure because we re duce the problem by invariance. We construct a sequence of cost func tions for which we obtain a sequence of solutions Bayesian which converges to the solution of the maximum likelihood. Results: We show that both solutions are asymptotically equal. Conclusion/Recommendation : The generalization when the parameter space is a local compact group.

Characterizable groups: Some results and open questions

Let X be an Abelian topological group and X∧ its dual group. A subgroup H of X is called characterized if there is a sequence {un} in X∧ such that H={x∈X:(un,x)→1}. A Polish Abelian group G is called characterizable if there is a continuous monomorphism p from G into a compact metrizable Abelian group X with dense image such that p(G) is a characterized subgroup of X. Every characterizable group is locally quasi-convex. We prove that every second countable locally compact Abelian group X is characterizable. Thus, every second countable locally compact Abelian group is the dual group of a complete countable maximally almost periodic group. It is shown that each characterizable Abelian group of finite exponent is locally compact. Analogously to the Abelian case, we define characterized subgroups of non-Abelian compact metrizable groups and non-Abelian characterizable groups. Using the ℓp-sum of metric groups with two-sided invariant metrics, it is proved that every characterized subgroup admits a Polish group topology.

The homogeneous space [formula omitted] as an equivariant fibrant space

In this paper we discuss some properties of equivariant fibrant spaces. It is shown that for every compact metrizable group G and its closed subgroup H, the quotient space G / H is a fibrant G-space.

On the existence of ergodic automorphisms in ergodic ℤd-actions on compact groups

AbstractLet K be a compact metrizable group and Γ be a finitely generated group of commuting automorphisms of K. We show that ergodicity of Γ implies Γ contains ergodic automorphisms if center of the action, Z(Γ)={α∈Aut(K)∣α commutes with elements of Γ} has descending chain condition. To explain that the condition on the center of the action is not restrictive, we discuss certain abelian groups which, in particular, provide new proofs to the theorems of Berend [Ergodic semigroups of epimorphisms. Trans. Amer. Math. Soc.289(1) (1985), 393–407] and Schmidt [Automorphisms of compact abelian groups and affine varieties. Proc. London Math. Soc. (3) 61 (1990), 480–496].

Some geometric and dynamical properties of the Urysohn space

This is a survey article about the geometry and dynamical properties of the Urysohn space. Most of the results presented here are part of the author's Ph.D. thesis and were published in the articles [J. Melleray, Stabilizers of closed sets in the Urysohn space, Fund. Math. 189 (1) (2006) 53–60; J. Melleray, Compact metrizable groups are isometry groups of compact metric spaces, Proc. Amer. Math. Soc. 136 (4) (2008) 1451; J. Melleray, On the geometry of Urysohn's universal metric space, Topology Appl. 154 (2007) 384–403]; a few results are new, most notably the fact that Iso ( U ) is not divisible.

Group shifts and Bernoulli factors

AbstractIn this paper, a group shift is an expansive action of $\Z ^d$ on a compact metrizable zero-dimensional group by continuous automorphisms. All group shifts factor topologically onto equal-entropy Bernoulli shifts; abelian group shifts factor by continuous group homomorphisms onto canonical equal-entropy Bernoulli group shifts; and completely positive entropy abelian group shifts are weakly algebraically equivalent to these Bernoulli factors. A completely positive entropy group (even vector) shift need not be topologically conjugate to a Bernoulli shift, and the Pinsker factor of a vector shift need not split topologically.

Compact metrizable groups are isometry groups of compact metric spaces

This note is devoted to proving the following result: given a compact metrizable group G, there is a compact metric space K such that G is isomorphic (as a topological group) to the isometry group of K.

Unitary representations and modular actions

We call a measure-preserving action of a countable discrete group on a standard probability space tempered if the associated Koopman representation restricted to the orthogonal complement to the constant functions is weakly contained in the regular representation. Extending a result of Hjorth, we show that every tempered action is antimodular, i.e., in a precise sense “orthogonal” to any Borel action of a countable group by automorphisms on a countable rooted tree. We also study tempered actions of countable groups by automorphisms on compact metrizable groups, where it turns out that this notion has several ergodic theoretic reformulations and fits naturally in a hierarchy of strong ergodicity properties strictly between ergodicity and strong mixing. Bibliography:s 25 titles.

Equivariant fibrant spaces

In this paper the concept of a G-bran t space is intro- duced. It is shown that any compact metrizable group G is a G-bran t.