{"title":"可数近似群渐近维度的 Hurewicz 和 Dranishnikov-Smith 定理","authors":"Tobias Hartnick , Vera Tonić","doi":"10.1016/j.topol.2024.108905","DOIUrl":null,"url":null,"abstract":"<div><p>We establish two main results for the asymptotic dimension of countable approximate groups. The first one is a Hurewicz type formula for a global morphism of countable approximate groups <span><math><mi>f</mi><mo>:</mo><mo>(</mo><mi>Ξ</mi><mo>,</mo><msup><mrow><mi>Ξ</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>)</mo><mo>→</mo><mo>(</mo><mi>Λ</mi><mo>,</mo><msup><mrow><mi>Λ</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>)</mo></math></span>, stating that <span><math><mi>asdim</mi><mspace></mspace><mi>Ξ</mi><mo>≤</mo><mi>asdim</mi><mspace></mspace><mi>Λ</mi><mo>+</mo><mi>asdim</mi><mo>(</mo><msub><mrow><mo>[</mo><mi>ker</mi><mo></mo><mi>f</mi><mo>]</mo></mrow><mrow><mi>c</mi></mrow></msub><mo>)</mo></math></span>. This is analogous to the Dranishnikov-Smith result for groups, and is relying on another Hurewicz type formula we prove, using a 6-local morphism instead of a global one. The second result is similar to the Dranishnikov-Smith theorem stating that, for a countable group <em>G</em>, asdim <em>G</em> is equal to the supremum of asymptotic dimensions of finitely generated subgroups of <em>G</em>. Our version states that, if <span><math><mo>(</mo><mi>Λ</mi><mo>,</mo><msup><mrow><mi>Λ</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>)</mo></math></span> is a countable approximate group, then asdim Λ is equal to the supremum of asymptotic dimensions of approximate subgroups of finitely generated subgroups of <span><math><msup><mrow><mi>Λ</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span>, with these approximate subgroups contained in <span><math><msup><mrow><mi>Λ</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Hurewicz and Dranishnikov-Smith theorems for asymptotic dimension of countable approximate groups\",\"authors\":\"Tobias Hartnick , Vera Tonić\",\"doi\":\"10.1016/j.topol.2024.108905\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We establish two main results for the asymptotic dimension of countable approximate groups. The first one is a Hurewicz type formula for a global morphism of countable approximate groups <span><math><mi>f</mi><mo>:</mo><mo>(</mo><mi>Ξ</mi><mo>,</mo><msup><mrow><mi>Ξ</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>)</mo><mo>→</mo><mo>(</mo><mi>Λ</mi><mo>,</mo><msup><mrow><mi>Λ</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>)</mo></math></span>, stating that <span><math><mi>asdim</mi><mspace></mspace><mi>Ξ</mi><mo>≤</mo><mi>asdim</mi><mspace></mspace><mi>Λ</mi><mo>+</mo><mi>asdim</mi><mo>(</mo><msub><mrow><mo>[</mo><mi>ker</mi><mo></mo><mi>f</mi><mo>]</mo></mrow><mrow><mi>c</mi></mrow></msub><mo>)</mo></math></span>. This is analogous to the Dranishnikov-Smith result for groups, and is relying on another Hurewicz type formula we prove, using a 6-local morphism instead of a global one. The second result is similar to the Dranishnikov-Smith theorem stating that, for a countable group <em>G</em>, asdim <em>G</em> is equal to the supremum of asymptotic dimensions of finitely generated subgroups of <em>G</em>. Our version states that, if <span><math><mo>(</mo><mi>Λ</mi><mo>,</mo><msup><mrow><mi>Λ</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>)</mo></math></span> is a countable approximate group, then asdim Λ is equal to the supremum of asymptotic dimensions of approximate subgroups of finitely generated subgroups of <span><math><msup><mrow><mi>Λ</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span>, with these approximate subgroups contained in <span><math><msup><mrow><mi>Λ</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>.</p></div>\",\"PeriodicalId\":51201,\"journal\":{\"name\":\"Topology and its Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-04-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Topology and its Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0166864124000907\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Topology and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0166864124000907","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
我们建立了可数近似群渐近维的两个主要结果。第一个是可数近似群的全局态 f:(Ξ,Ξ∞)→(Λ,Λ∞) 的胡勒维茨式公式,指出 asdimΞ≤asdimΛ+asdim([kerf]c) 。这与群的德拉尼什尼科夫-史密斯结果类似,并且依赖于我们证明的另一个胡勒维茨式,使用的是 6 局部态而不是全局态。第二个结果类似于德拉尼什尼科夫-史密斯定理,即对于可数群 G,asdim G 等于 G 的有限生成子群的渐近维数的上集。我们的版本指出,如果 (Λ,Λ∞)是一个可数近似群,那么 asdim Λ 等于 Λ∞ 的有限生成子群的近似子群的渐近维数的上集,这些近似子群包含在Λ2 中。
Hurewicz and Dranishnikov-Smith theorems for asymptotic dimension of countable approximate groups
We establish two main results for the asymptotic dimension of countable approximate groups. The first one is a Hurewicz type formula for a global morphism of countable approximate groups , stating that . This is analogous to the Dranishnikov-Smith result for groups, and is relying on another Hurewicz type formula we prove, using a 6-local morphism instead of a global one. The second result is similar to the Dranishnikov-Smith theorem stating that, for a countable group G, asdim G is equal to the supremum of asymptotic dimensions of finitely generated subgroups of G. Our version states that, if is a countable approximate group, then asdim Λ is equal to the supremum of asymptotic dimensions of approximate subgroups of finitely generated subgroups of , with these approximate subgroups contained in .
期刊介绍:
Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology.
At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.