Asymptotic and Assouad–Nagata dimension of finitely generated groups and their subgroups

Pub Date : 2023-12-03 DOI:10.1112/topo.12314

Levi Sledd

{"title":"Asymptotic and Assouad–Nagata dimension of finitely generated groups and their subgroups","authors":"Levi Sledd","doi":"10.1112/topo.12314","DOIUrl":null,"url":null,"abstract":"<p>We prove that for all <math>\n <semantics>\n <mrow>\n <mi>k</mi>\n <mo>,</mo>\n <mi>m</mi>\n <mo>,</mo>\n <mi>n</mi>\n <mo>∈</mo>\n <mi>N</mi>\n <mo>∪</mo>\n <mo>{</mo>\n <mi>∞</mi>\n <mo>}</mo>\n </mrow>\n <annotation>$k,m,n \\in \\mathbb {N} \\cup \\lbrace \\infty \\rbrace$</annotation>\n </semantics></math> with <math>\n <semantics>\n <mrow>\n <mn>4</mn>\n <mo>⩽</mo>\n <mi>k</mi>\n <mo>⩽</mo>\n <mi>m</mi>\n <mo>⩽</mo>\n <mi>n</mi>\n </mrow>\n <annotation>$4 \\leqslant k \\leqslant m \\leqslant n$</annotation>\n </semantics></math>, there exists a finitely generated group <math>\n <semantics>\n <mi>G</mi>\n <annotation>$G$</annotation>\n </semantics></math> with a finitely generated subgroup <math>\n <semantics>\n <mi>H</mi>\n <annotation>$H$</annotation>\n </semantics></math> such that <math>\n <semantics>\n <mrow>\n <mo>asdim</mo>\n <mo>(</mo>\n <mi>G</mi>\n <mo>)</mo>\n <mo>=</mo>\n <mi>k</mi>\n </mrow>\n <annotation>$\\operatorname{asdim}(G) = k$</annotation>\n </semantics></math>, <math>\n <semantics>\n <mrow>\n <msub>\n <mo>asdim</mo>\n <mrow>\n <mi>A</mi>\n <mi>N</mi>\n </mrow>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>G</mi>\n <mo>)</mo>\n </mrow>\n <mo>=</mo>\n <mi>m</mi>\n </mrow>\n <annotation>$\\operatorname{asdim}_{\\textnormal {AN}}(G) = m$</annotation>\n </semantics></math>, and <math>\n <semantics>\n <mrow>\n <msub>\n <mo>asdim</mo>\n <mrow>\n <mi>A</mi>\n <mi>N</mi>\n </mrow>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>H</mi>\n <mo>)</mo>\n </mrow>\n <mo>=</mo>\n <mi>n</mi>\n </mrow>\n <annotation>$\\operatorname{asdim}_{\\textnormal {AN}}(H)=n$</annotation>\n </semantics></math>. This simultaneously answers two open questions in asymptotic dimension theory.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/topo.12314","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 1

Abstract

We prove that for all $k, m, n \in N \cup {\infty}$ with $4 ⩽ k ⩽ m ⩽ n$ , there exists a finitely generated group $G$ with a finitely generated subgroup $H$ such that $asdim (G) = k$ , ${asdim}_{A N} (G) = m$ , and ${asdim}_{A N} (H) = n$ . This simultaneously answers two open questions in asymptotic dimension theory.

查看原文

有限生成群及其子群的渐近维数和副长维数

我们向所有k m证明，n ∈ N ∪ { ∞ } $ k, m, n在杯赛mathbb {n} \ \ lbrace infty \ rbrace $ 一起散步 4 ⩽ k ⩽ m ⩽ n $ 4 \ leqslant k leqslant m \ leqslant n $ ,在一个有限的G美元集团中存在着一个有限的G美元子集团H美元H这样的asdim (G)asdim AN (G)和asdim AN (H)这实际上是两个在异步维度问题的答案。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文