{"title":"Nonfree almost finite actions for locally finite-by-virtually \n \n Z\n ${\\mathbb {Z}}$\n groups","authors":"Kang Li, Xin Ma","doi":"10.1112/jlms.12959","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we study almost finiteness and almost finiteness in measure of nonfree actions. Let <span></span><math>\n <semantics>\n <mrow>\n <mi>α</mi>\n <mo>:</mo>\n <mi>G</mi>\n <mi>↷</mi>\n <mi>X</mi>\n </mrow>\n <annotation>$\\alpha:G\\curvearrowright X$</annotation>\n </semantics></math> be a minimal action of a locally finite-by-virtually <span></span><math>\n <semantics>\n <mi>Z</mi>\n <annotation>${\\mathbb {Z}}$</annotation>\n </semantics></math> group <span></span><math>\n <semantics>\n <mi>G</mi>\n <annotation>$G$</annotation>\n </semantics></math> on the Cantor set <span></span><math>\n <semantics>\n <mi>X</mi>\n <annotation>$X$</annotation>\n </semantics></math>. We prove that under certain assumptions, the action <span></span><math>\n <semantics>\n <mi>α</mi>\n <annotation>$\\alpha$</annotation>\n </semantics></math> is almost finite in measure if and only if <span></span><math>\n <semantics>\n <mi>α</mi>\n <annotation>$\\alpha$</annotation>\n </semantics></math> is essentially free. As an application, we obtain that any minimal topologically free action of a virtually <span></span><math>\n <semantics>\n <mi>Z</mi>\n <annotation>${\\mathbb {Z}}$</annotation>\n </semantics></math> group on an infinite compact metrizable space with the small boundary property is almost finite. This is the first general result, assuming only topological freeness, in this direction, and these lead to new results on uniform property <span></span><math>\n <semantics>\n <mi>Γ</mi>\n <annotation>$\\Gamma$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mi>Z</mi>\n <annotation>$\\mathcal {Z}$</annotation>\n </semantics></math>-stability for their crossed product <span></span><math>\n <semantics>\n <msup>\n <mi>C</mi>\n <mo>∗</mo>\n </msup>\n <annotation>$C^*$</annotation>\n </semantics></math>-algebras. Some concrete examples of minimal topological free (but nonfree) subshifts are provided.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12959","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we study almost finiteness and almost finiteness in measure of nonfree actions. Let be a minimal action of a locally finite-by-virtually group on the Cantor set . We prove that under certain assumptions, the action is almost finite in measure if and only if is essentially free. As an application, we obtain that any minimal topologically free action of a virtually group on an infinite compact metrizable space with the small boundary property is almost finite. This is the first general result, assuming only topological freeness, in this direction, and these lead to new results on uniform property and -stability for their crossed product -algebras. Some concrete examples of minimal topological free (but nonfree) subshifts are provided.
期刊介绍:
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
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