{"title":"Groups with Spanier-Whitehead Duality","authors":"Shintaro Nishikawa, Valerio Proietti","doi":"10.14760/OWP-2019-23","DOIUrl":null,"url":null,"abstract":"We introduce the notion of Spanier-Whitehead K-duality for a discrete group G, defined as duality in the KK-category between two C*-algebras which are naturally attached to the group, namely the reduced group C*-algebra and the crossed product for the group action on the universal example for proper actions. We compare this notion to the Baum-Connes conjecture by constructing duality classes based on two methods: the standard \"gamma element\" technique, and the more recent approach via cycles with property gamma. As a result of our analysis, we prove Spanier-Whitehead duality for a large class of groups, including Bieberbach's space groups, groups acting on trees, and lattices in Lorentz groups.","PeriodicalId":309711,"journal":{"name":"arXiv: K-Theory and Homology","volume":"27 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2019-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"6","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: K-Theory and Homology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.14760/OWP-2019-23","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 6
Abstract
We introduce the notion of Spanier-Whitehead K-duality for a discrete group G, defined as duality in the KK-category between two C*-algebras which are naturally attached to the group, namely the reduced group C*-algebra and the crossed product for the group action on the universal example for proper actions. We compare this notion to the Baum-Connes conjecture by constructing duality classes based on two methods: the standard "gamma element" technique, and the more recent approach via cycles with property gamma. As a result of our analysis, we prove Spanier-Whitehead duality for a large class of groups, including Bieberbach's space groups, groups acting on trees, and lattices in Lorentz groups.