Pub Date : 2024-03-11DOI: 10.1017/s0017089524000065
Qizheng You, Jiawen Zhang
In this note, we present examples of non-quasi-geodesic metric spaces which are hyperbolic (i.e., satisfying Gromov’s $4$-point condition) while the intersection of any two metric balls therein does not either ‘look like’ a ball or has uniformly bounded eccentricity. This answers an open question posed by Chatterji and Niblo.
{"title":"Examples of hyperbolic spaces without the properties of quasi-ball or bounded eccentricity","authors":"Qizheng You, Jiawen Zhang","doi":"10.1017/s0017089524000065","DOIUrl":"https://doi.org/10.1017/s0017089524000065","url":null,"abstract":"<p>In this note, we present examples of non-quasi-geodesic metric spaces which are hyperbolic (i.e., satisfying Gromov’s <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240308121158651-0633:S0017089524000065:S0017089524000065_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$4$</span></span></img></span></span>-point condition) while the intersection of any two metric balls therein does not either ‘look like’ a ball or has uniformly bounded eccentricity. This answers an open question posed by Chatterji and Niblo.</p>","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"127 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140097905","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-16DOI: 10.1017/s0017089524000041
Gurleen Kaur, Surinder Kaur, Pooja Singla
In this article, we explore the problem of determining isomorphisms between the twisted complex group algebras of finite $p$ -groups. This problem bears similarity to the classical group algebra isomorphism problem and has been recently examined by Margolis-Schnabel. Our focus lies on a specific invariant, referred to as the generalized corank, which relates to the twisted complex group algebra isomorphism problem. We provide a solution for non-abelian $p$ -groups with generalized corank at most three.
{"title":"On twisted group ring isomorphism problem for p-groups","authors":"Gurleen Kaur, Surinder Kaur, Pooja Singla","doi":"10.1017/s0017089524000041","DOIUrl":"https://doi.org/10.1017/s0017089524000041","url":null,"abstract":"In this article, we explore the problem of determining isomorphisms between the twisted complex group algebras of finite <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000041_inline1.png\" /> <jats:tex-math> $p$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-groups. This problem bears similarity to the classical group algebra isomorphism problem and has been recently examined by Margolis-Schnabel. Our focus lies on a specific invariant, referred to as the generalized corank, which relates to the twisted complex group algebra isomorphism problem. We provide a solution for non-abelian <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000041_inline2.png\" /> <jats:tex-math> $p$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-groups with generalized corank at most three.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"11 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139766048","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-02DOI: 10.1017/s0017089524000028
Moshe Jarden
Generalizing a result of Wulf-Dieter Geyer in his thesis, we prove that if $K$ is a finitely generated extension of transcendence degree $r$ of a global field and $A$ is a closed abelian subgroup of $textrm{Gal}(K)$ , then ${mathrm{rank}}(A)le r+1$ . Moreover, if $mathrm{char}(K)=0$ , then ${hat{mathbb{Z}}}^{r+1}$ is isomorphic to a closed subgroup of $textrm{Gal}(K)$ .
Pub Date : 2024-01-26DOI: 10.1017/s0017089524000016
Katharina Müller
In this paper, we prove Kato’s main conjecture for $CM$ modular forms for primes of potentially ordinary reduction under certain hypotheses on the modular form.
{"title":"Kato’s main conjecture for potentially ordinary primes","authors":"Katharina Müller","doi":"10.1017/s0017089524000016","DOIUrl":"https://doi.org/10.1017/s0017089524000016","url":null,"abstract":"In this paper, we prove Kato’s main conjecture for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089524000016_inline1.png\" /> <jats:tex-math> $CM$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> modular forms for primes of potentially ordinary reduction under certain hypotheses on the modular form.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"8 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139582750","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-22DOI: 10.1017/s0017089523000460
Fabio Ferri
<p>We prove new results concerning the additive Galois module structure of wildly ramified non-abelian extensions <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240121224918357-0465:S0017089523000460:S0017089523000460_inline2.png"><span data-mathjax-type="texmath"><span>$K/mathbb{Q}$</span></span></img></span></span> with Galois group isomorphic to <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240121224918357-0465:S0017089523000460:S0017089523000460_inline3.png"><span data-mathjax-type="texmath"><span>$A_4$</span></span></img></span></span>, <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240121224918357-0465:S0017089523000460:S0017089523000460_inline4.png"><span data-mathjax-type="texmath"><span>$S_4$</span></span></img></span></span>, <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240121224918357-0465:S0017089523000460:S0017089523000460_inline5.png"><span data-mathjax-type="texmath"><span>$A_5$</span></span></img></span></span>, and dihedral groups of order <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240121224918357-0465:S0017089523000460:S0017089523000460_inline6.png"><span data-mathjax-type="texmath"><span>$2p^n$</span></span></img></span></span> for certain prime powers <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240121224918357-0465:S0017089523000460:S0017089523000460_inline7.png"><span data-mathjax-type="texmath"><span>$p^n$</span></span></img></span></span>. In particular, when <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240121224918357-0465:S0017089523000460:S0017089523000460_inline8.png"><span data-mathjax-type="texmath"><span>$K/mathbb{Q}$</span></span></img></span></span> is a Galois extension with Galois group <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240121224918357-0465:S0017089523000460:S0017089523000460_inline9.png"><span data-mathjax-type="texmath"><span>$G$</span></span></img></span></span> isomorphic to <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240121224918357-0465:S0017089523000460:S0017089523000460_inline10.png"><span data-mathjax-type="texmath"><span>$A_4$</span></span></img></span></span>, <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:2024012
{"title":"Leopoldt-type theorems for non-abelian extensions of","authors":"Fabio Ferri","doi":"10.1017/s0017089523000460","DOIUrl":"https://doi.org/10.1017/s0017089523000460","url":null,"abstract":"<p>We prove new results concerning the additive Galois module structure of wildly ramified non-abelian extensions <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240121224918357-0465:S0017089523000460:S0017089523000460_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$K/mathbb{Q}$</span></span></img></span></span> with Galois group isomorphic to <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240121224918357-0465:S0017089523000460:S0017089523000460_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$A_4$</span></span></img></span></span>, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240121224918357-0465:S0017089523000460:S0017089523000460_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$S_4$</span></span></img></span></span>, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240121224918357-0465:S0017089523000460:S0017089523000460_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$A_5$</span></span></img></span></span>, and dihedral groups of order <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240121224918357-0465:S0017089523000460:S0017089523000460_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$2p^n$</span></span></img></span></span> for certain prime powers <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240121224918357-0465:S0017089523000460:S0017089523000460_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$p^n$</span></span></img></span></span>. In particular, when <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240121224918357-0465:S0017089523000460:S0017089523000460_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$K/mathbb{Q}$</span></span></img></span></span> is a Galois extension with Galois group <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240121224918357-0465:S0017089523000460:S0017089523000460_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$G$</span></span></img></span></span> isomorphic to <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240121224918357-0465:S0017089523000460:S0017089523000460_inline10.png\"><span data-mathjax-type=\"texmath\"><span>$A_4$</span></span></img></span></span>, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:2024012","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"38 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139517126","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-11DOI: 10.1017/s0017089523000459
Kristen Pueschel, Timothy Riley
<p>The algebraic mapping torus <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110051631416-0480:S0017089523000459:S0017089523000459_inline1.png"><span data-mathjax-type="texmath"><span>$M_{Phi }$</span></span></img></span></span> of a group <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110051631416-0480:S0017089523000459:S0017089523000459_inline2.png"><span data-mathjax-type="texmath"><span>$G$</span></span></img></span></span> with an automorphism <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110051631416-0480:S0017089523000459:S0017089523000459_inline3.png"><span data-mathjax-type="texmath"><span>$Phi$</span></span></img></span></span> is the HNN-extension of <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110051631416-0480:S0017089523000459:S0017089523000459_inline4.png"><span data-mathjax-type="texmath"><span>$G$</span></span></img></span></span> in which conjugation by the stable letter performs <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110051631416-0480:S0017089523000459:S0017089523000459_inline5.png"><span data-mathjax-type="texmath"><span>$Phi$</span></span></img></span></span>. We classify the Dehn functions of <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110051631416-0480:S0017089523000459:S0017089523000459_inline6.png"><span data-mathjax-type="texmath"><span>$M_{Phi }$</span></span></img></span></span> in terms of <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110051631416-0480:S0017089523000459:S0017089523000459_inline7.png"><span data-mathjax-type="texmath"><span>$Phi$</span></span></img></span></span> for a number of right-angled Artin groups (RAAGs) <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110051631416-0480:S0017089523000459:S0017089523000459_inline8.png"><span data-mathjax-type="texmath"><span>$G$</span></span></img></span></span>, including all <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110051631416-0480:S0017089523000459:S0017089523000459_inline9.png"><span data-mathjax-type="texmath"><span>$3$</span></span></img></span></span>-generator RAAGs and <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110051631416-0480:S001708
{"title":"Dehn functions of mapping tori of right-angled Artin groups","authors":"Kristen Pueschel, Timothy Riley","doi":"10.1017/s0017089523000459","DOIUrl":"https://doi.org/10.1017/s0017089523000459","url":null,"abstract":"<p>The algebraic mapping torus <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110051631416-0480:S0017089523000459:S0017089523000459_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$M_{Phi }$</span></span></img></span></span> of a group <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110051631416-0480:S0017089523000459:S0017089523000459_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$G$</span></span></img></span></span> with an automorphism <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110051631416-0480:S0017089523000459:S0017089523000459_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$Phi$</span></span></img></span></span> is the HNN-extension of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110051631416-0480:S0017089523000459:S0017089523000459_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$G$</span></span></img></span></span> in which conjugation by the stable letter performs <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110051631416-0480:S0017089523000459:S0017089523000459_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$Phi$</span></span></img></span></span>. We classify the Dehn functions of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110051631416-0480:S0017089523000459:S0017089523000459_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$M_{Phi }$</span></span></img></span></span> in terms of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110051631416-0480:S0017089523000459:S0017089523000459_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$Phi$</span></span></img></span></span> for a number of right-angled Artin groups (RAAGs) <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110051631416-0480:S0017089523000459:S0017089523000459_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$G$</span></span></img></span></span>, including all <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110051631416-0480:S0017089523000459:S0017089523000459_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$3$</span></span></img></span></span>-generator RAAGs and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110051631416-0480:S001708","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"115 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139423118","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-11DOI: 10.1017/s0017089523000496
Porfirio L. León Álvarez
<p>Given a group <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110064323589-0660:S0017089523000496:S0017089523000496_inline1.png"><span data-mathjax-type="texmath"><span>$G$</span></span></img></span></span> and an integer <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110064323589-0660:S0017089523000496:S0017089523000496_inline2.png"><span data-mathjax-type="texmath"><span>$ngeq 0$</span></span></img></span></span>, we consider the family <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110064323589-0660:S0017089523000496:S0017089523000496_inline3.png"><span data-mathjax-type="texmath"><span>${mathcal F}_n$</span></span></img></span></span> of all virtually abelian subgroups of <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110064323589-0660:S0017089523000496:S0017089523000496_inline4.png"><span data-mathjax-type="texmath"><span>$G$</span></span></img></span></span> of <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110064323589-0660:S0017089523000496:S0017089523000496_inline5.png"><span data-mathjax-type="texmath"><span>$textrm{rank}$</span></span></img></span></span> at most <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110064323589-0660:S0017089523000496:S0017089523000496_inline6.png"><span data-mathjax-type="texmath"><span>$n$</span></span></img></span></span>. In this article, we prove that for each <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110064323589-0660:S0017089523000496:S0017089523000496_inline7.png"><span data-mathjax-type="texmath"><span>$nge 2$</span></span></img></span></span> the Bredon cohomology, with respect to the family <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110064323589-0660:S0017089523000496:S0017089523000496_inline8.png"><span data-mathjax-type="texmath"><span>${mathcal F}_n$</span></span></img></span></span>, of a free abelian group with <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110064323589-0660:S0017089523000496:S0017089523000496_inline9.png"><span data-mathjax-type="texmath"><span>$textrm{rank}$</span></span></img></span></span> <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110064323589-0660:S001708952300049
{"title":"Classifying spaces for families of abelian subgroups of braid groups, RAAGs and graphs of abelian groups","authors":"Porfirio L. León Álvarez","doi":"10.1017/s0017089523000496","DOIUrl":"https://doi.org/10.1017/s0017089523000496","url":null,"abstract":"<p>Given a group <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110064323589-0660:S0017089523000496:S0017089523000496_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$G$</span></span></img></span></span> and an integer <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110064323589-0660:S0017089523000496:S0017089523000496_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$ngeq 0$</span></span></img></span></span>, we consider the family <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110064323589-0660:S0017089523000496:S0017089523000496_inline3.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal F}_n$</span></span></img></span></span> of all virtually abelian subgroups of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110064323589-0660:S0017089523000496:S0017089523000496_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$G$</span></span></img></span></span> of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110064323589-0660:S0017089523000496:S0017089523000496_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$textrm{rank}$</span></span></img></span></span> at most <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110064323589-0660:S0017089523000496:S0017089523000496_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$n$</span></span></img></span></span>. In this article, we prove that for each <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110064323589-0660:S0017089523000496:S0017089523000496_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$nge 2$</span></span></img></span></span> the Bredon cohomology, with respect to the family <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110064323589-0660:S0017089523000496:S0017089523000496_inline8.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal F}_n$</span></span></img></span></span>, of a free abelian group with <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110064323589-0660:S0017089523000496:S0017089523000496_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$textrm{rank}$</span></span></img></span></span> <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240110064323589-0660:S001708952300049","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"8 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139423921","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-09DOI: 10.1017/s0017089523000435
C.R. Leedham-Green
We discuss a variant, named ‘Rattle’, of the product replacement algorithm. Rattle is a Markov chain, that returns a random element of a black box group. The limiting distribution of the element returned is the uniform distribution. We prove that, if the generating sequence is long enough, the probability distribution of the element returned converges unexpectedly quickly to the uniform distribution.
{"title":"On a variant of the product replacement algorithm","authors":"C.R. Leedham-Green","doi":"10.1017/s0017089523000435","DOIUrl":"https://doi.org/10.1017/s0017089523000435","url":null,"abstract":"We discuss a variant, named ‘Rattle’, of the product replacement algorithm. Rattle is a Markov chain, that returns a random element of a black box group. The limiting distribution of the element returned is the uniform distribution. We prove that, if the generating sequence is long enough, the probability distribution of the element returned converges unexpectedly quickly to the uniform distribution.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"20 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139410155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-15DOI: 10.1017/s0017089523000447
Pawel Sarkowicz
We show that continuous group homomorphisms between unitary groups of unital C*-algebras induce maps between spaces of continuous real-valued affine functions on the trace simplices. Under certain $K$ -theoretic regularity conditions, these maps can be seen to commute with the pairing between $K_0$ and traces. If the homomorphism is contractive and sends the unit circle to the unit circle, the map between spaces of continuous real-valued affine functions can further be shown to be unital and positive (up to a minus sign).
{"title":"Unitary groups, -theory, and traces","authors":"Pawel Sarkowicz","doi":"10.1017/s0017089523000447","DOIUrl":"https://doi.org/10.1017/s0017089523000447","url":null,"abstract":"We show that continuous group homomorphisms between unitary groups of unital C*-algebras induce maps between spaces of continuous real-valued affine functions on the trace simplices. Under certain <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089523000447_inline2.png\" /> <jats:tex-math> $K$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-theoretic regularity conditions, these maps can be seen to commute with the pairing between <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0017089523000447_inline3.png\" /> <jats:tex-math> $K_0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and traces. If the homomorphism is contractive and sends the unit circle to the unit circle, the map between spaces of continuous real-valued affine functions can further be shown to be unital and positive (up to a minus sign).","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"298 1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138691449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-11DOI: 10.1017/s0017089523000423
Donald M. Davis, W. Stephen Wilson
<p>We compute <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207164304020-0554:S0017089523000423:S0017089523000423_inline3.png"><span data-mathjax-type="texmath"><span>$ku^*left(K!left({mathbb{Z}}_p,2right)right)$</span></span></img></span></span> and <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207164304020-0554:S0017089523000423:S0017089523000423_inline4.png"><span data-mathjax-type="texmath"><span>$ku_*left(K!left({mathbb{Z}}_p,2right)right)$</span></span></img></span></span>, the connective <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207164304020-0554:S0017089523000423:S0017089523000423_inline5.png"><span data-mathjax-type="texmath"><span>$KU$</span></span></img></span></span>-cohomology and connective <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207164304020-0554:S0017089523000423:S0017089523000423_inline6.png"><span data-mathjax-type="texmath"><span>$KU$</span></span></img></span></span>-homology groups of the mod-<span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207164304020-0554:S0017089523000423:S0017089523000423_inline7.png"><span data-mathjax-type="texmath"><span>$p$</span></span></img></span></span> Eilenberg–MacLane space <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207164304020-0554:S0017089523000423:S0017089523000423_inline8.png"><span data-mathjax-type="texmath"><span>$K!left({mathbb{Z}}_p,2right)$</span></span></img></span></span>, using the Adams spectral sequence. We obtain a striking interaction between <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207164304020-0554:S0017089523000423:S0017089523000423_inline9.png"><span data-mathjax-type="texmath"><span>$h_0$</span></span></img></span></span>-extensions and exotic extensions. The mod-<span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207164304020-0554:S0017089523000423:S0017089523000423_inline10.png"><span data-mathjax-type="texmath"><span>$p$</span></span></img></span></span> connective <span><span><img data-mimesubtype="png" data-type="" src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207164304020-0554:S0017089523000423:S0017089523000423_inline11.png"><span data-mathjax-type="texmath"><span>$KU$</span></span></img></span></span>-cohomology groups, computed elsewhere, are needed in order to establish higher differentials a
{"title":"The connective","authors":"Donald M. Davis, W. Stephen Wilson","doi":"10.1017/s0017089523000423","DOIUrl":"https://doi.org/10.1017/s0017089523000423","url":null,"abstract":"<p>We compute <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207164304020-0554:S0017089523000423:S0017089523000423_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$ku^*left(K!left({mathbb{Z}}_p,2right)right)$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207164304020-0554:S0017089523000423:S0017089523000423_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$ku_*left(K!left({mathbb{Z}}_p,2right)right)$</span></span></img></span></span>, the connective <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207164304020-0554:S0017089523000423:S0017089523000423_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$KU$</span></span></img></span></span>-cohomology and connective <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207164304020-0554:S0017089523000423:S0017089523000423_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$KU$</span></span></img></span></span>-homology groups of the mod-<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207164304020-0554:S0017089523000423:S0017089523000423_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$p$</span></span></img></span></span> Eilenberg–MacLane space <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207164304020-0554:S0017089523000423:S0017089523000423_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$K!left({mathbb{Z}}_p,2right)$</span></span></img></span></span>, using the Adams spectral sequence. We obtain a striking interaction between <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207164304020-0554:S0017089523000423:S0017089523000423_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$h_0$</span></span></img></span></span>-extensions and exotic extensions. The mod-<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207164304020-0554:S0017089523000423:S0017089523000423_inline10.png\"><span data-mathjax-type=\"texmath\"><span>$p$</span></span></img></span></span> connective <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231207164304020-0554:S0017089523000423:S0017089523000423_inline11.png\"><span data-mathjax-type=\"texmath\"><span>$KU$</span></span></img></span></span>-cohomology groups, computed elsewhere, are needed in order to establish higher differentials a","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"230 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138569427","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
$4$-point condition) while the intersection of any two metric balls therein does not either ‘look like’ a ball or has uniformly bounded eccentricity. This answers an open question posed by Chatterji and Niblo.
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号