Dimension drop in residual chains

We give a description of the Linnell division ring of a countable residually (poly-$\mathbb Z$ virtually nilpotent) (RPVN) group in terms of a generalised Novikov ring, and show that vanishing top-degree cohomology of a finite type group $G$ with coefficients in this Novikov ring implies the existen...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Hauptverfasser: Fisher, Sam P, Klinge, Kevin
Format: Artikel
Sprache:eng
Schlagworte:
Online-Zugang:Volltext bestellen
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
container_end_page
container_issue
container_start_page
container_title
container_volume
creator Fisher, Sam P
Klinge, Kevin
description We give a description of the Linnell division ring of a countable residually (poly-$\mathbb Z$ virtually nilpotent) (RPVN) group in terms of a generalised Novikov ring, and show that vanishing top-degree cohomology of a finite type group $G$ with coefficients in this Novikov ring implies the existence of a normal subgroup $N \leqslant G$ such that $\mathrm{cd}_{\mathbb Q}(N) < \mathrm{cd}_{\mathbb Q}(G)$ and $G/N$ is poly-$\mathbb Z$ virtually nilpotent. As a consequence, we show that if $G$ is an RPVN group of finite type, then its top-degree $\ell^2$-Betti number vanishes if and only if there is a poly-$\mathbb Z$ virtually nilpotent quotient $G/N$ such that $\mathrm{cd}_{\mathbb Q}(N) < \mathrm{cd}_{\mathbb Q}(G)$. In particular, finitely generated RPVN groups of cohomological dimension $2$ are virtually free-by-nilpotent if and only if their second $\ell^2$-Betti number vanishes, and therefore $2$-dimensional RPVN groups with vanishing second $\ell^2$-Betti number are coherent. As another application, we show that if $G$ is a finitely generated parafree group with $\mathrm{cd}(G) = 2$, then $G$ satisfies the Parafree Conjecture if and only if the terms of its lower central series are eventually free. Note that the class of RPVN groups contains all finitely generated RFRS groups and all finitely generated residually torsion-free nilpotent groups.
doi_str_mv 10.48550/arxiv.2410.08153
format Article
fullrecord <record><control><sourceid>arxiv_GOX</sourceid><recordid>TN_cdi_arxiv_primary_2410_08153</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2410_08153</sourcerecordid><originalsourceid>FETCH-arxiv_primary_2410_081533</originalsourceid><addsrcrecordid>eNpjYJA0NNAzsTA1NdBPLKrILNMzMgEKGFgYmhpzMii6ZOam5hVn5ucppBTlFyhk5ikUpRZnppQm5igkZyRm5hXzMLCmJeYUp_JCaW4GeTfXEGcPXbBh8QVFmbmJRZXxIEPjwYYaE1YBACj9Kx4</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>Dimension drop in residual chains</title><source>arXiv.org</source><creator>Fisher, Sam P ; Klinge, Kevin</creator><creatorcontrib>Fisher, Sam P ; Klinge, Kevin</creatorcontrib><description>We give a description of the Linnell division ring of a countable residually (poly-$\mathbb Z$ virtually nilpotent) (RPVN) group in terms of a generalised Novikov ring, and show that vanishing top-degree cohomology of a finite type group $G$ with coefficients in this Novikov ring implies the existence of a normal subgroup $N \leqslant G$ such that $\mathrm{cd}_{\mathbb Q}(N) &lt; \mathrm{cd}_{\mathbb Q}(G)$ and $G/N$ is poly-$\mathbb Z$ virtually nilpotent. As a consequence, we show that if $G$ is an RPVN group of finite type, then its top-degree $\ell^2$-Betti number vanishes if and only if there is a poly-$\mathbb Z$ virtually nilpotent quotient $G/N$ such that $\mathrm{cd}_{\mathbb Q}(N) &lt; \mathrm{cd}_{\mathbb Q}(G)$. In particular, finitely generated RPVN groups of cohomological dimension $2$ are virtually free-by-nilpotent if and only if their second $\ell^2$-Betti number vanishes, and therefore $2$-dimensional RPVN groups with vanishing second $\ell^2$-Betti number are coherent. As another application, we show that if $G$ is a finitely generated parafree group with $\mathrm{cd}(G) = 2$, then $G$ satisfies the Parafree Conjecture if and only if the terms of its lower central series are eventually free. Note that the class of RPVN groups contains all finitely generated RFRS groups and all finitely generated residually torsion-free nilpotent groups.</description><identifier>DOI: 10.48550/arxiv.2410.08153</identifier><language>eng</language><subject>Mathematics - Group Theory</subject><creationdate>2024-10</creationdate><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,780,885</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/2410.08153$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.2410.08153$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Fisher, Sam P</creatorcontrib><creatorcontrib>Klinge, Kevin</creatorcontrib><title>Dimension drop in residual chains</title><description>We give a description of the Linnell division ring of a countable residually (poly-$\mathbb Z$ virtually nilpotent) (RPVN) group in terms of a generalised Novikov ring, and show that vanishing top-degree cohomology of a finite type group $G$ with coefficients in this Novikov ring implies the existence of a normal subgroup $N \leqslant G$ such that $\mathrm{cd}_{\mathbb Q}(N) &lt; \mathrm{cd}_{\mathbb Q}(G)$ and $G/N$ is poly-$\mathbb Z$ virtually nilpotent. As a consequence, we show that if $G$ is an RPVN group of finite type, then its top-degree $\ell^2$-Betti number vanishes if and only if there is a poly-$\mathbb Z$ virtually nilpotent quotient $G/N$ such that $\mathrm{cd}_{\mathbb Q}(N) &lt; \mathrm{cd}_{\mathbb Q}(G)$. In particular, finitely generated RPVN groups of cohomological dimension $2$ are virtually free-by-nilpotent if and only if their second $\ell^2$-Betti number vanishes, and therefore $2$-dimensional RPVN groups with vanishing second $\ell^2$-Betti number are coherent. As another application, we show that if $G$ is a finitely generated parafree group with $\mathrm{cd}(G) = 2$, then $G$ satisfies the Parafree Conjecture if and only if the terms of its lower central series are eventually free. Note that the class of RPVN groups contains all finitely generated RFRS groups and all finitely generated residually torsion-free nilpotent groups.</description><subject>Mathematics - Group Theory</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNpjYJA0NNAzsTA1NdBPLKrILNMzMgEKGFgYmhpzMii6ZOam5hVn5ucppBTlFyhk5ikUpRZnppQm5igkZyRm5hXzMLCmJeYUp_JCaW4GeTfXEGcPXbBh8QVFmbmJRZXxIEPjwYYaE1YBACj9Kx4</recordid><startdate>20241010</startdate><enddate>20241010</enddate><creator>Fisher, Sam P</creator><creator>Klinge, Kevin</creator><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20241010</creationdate><title>Dimension drop in residual chains</title><author>Fisher, Sam P ; Klinge, Kevin</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-arxiv_primary_2410_081533</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Mathematics - Group Theory</topic><toplevel>online_resources</toplevel><creatorcontrib>Fisher, Sam P</creatorcontrib><creatorcontrib>Klinge, Kevin</creatorcontrib><collection>arXiv Mathematics</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Fisher, Sam P</au><au>Klinge, Kevin</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Dimension drop in residual chains</atitle><date>2024-10-10</date><risdate>2024</risdate><abstract>We give a description of the Linnell division ring of a countable residually (poly-$\mathbb Z$ virtually nilpotent) (RPVN) group in terms of a generalised Novikov ring, and show that vanishing top-degree cohomology of a finite type group $G$ with coefficients in this Novikov ring implies the existence of a normal subgroup $N \leqslant G$ such that $\mathrm{cd}_{\mathbb Q}(N) &lt; \mathrm{cd}_{\mathbb Q}(G)$ and $G/N$ is poly-$\mathbb Z$ virtually nilpotent. As a consequence, we show that if $G$ is an RPVN group of finite type, then its top-degree $\ell^2$-Betti number vanishes if and only if there is a poly-$\mathbb Z$ virtually nilpotent quotient $G/N$ such that $\mathrm{cd}_{\mathbb Q}(N) &lt; \mathrm{cd}_{\mathbb Q}(G)$. In particular, finitely generated RPVN groups of cohomological dimension $2$ are virtually free-by-nilpotent if and only if their second $\ell^2$-Betti number vanishes, and therefore $2$-dimensional RPVN groups with vanishing second $\ell^2$-Betti number are coherent. As another application, we show that if $G$ is a finitely generated parafree group with $\mathrm{cd}(G) = 2$, then $G$ satisfies the Parafree Conjecture if and only if the terms of its lower central series are eventually free. Note that the class of RPVN groups contains all finitely generated RFRS groups and all finitely generated residually torsion-free nilpotent groups.</abstract><doi>10.48550/arxiv.2410.08153</doi><oa>free_for_read</oa></addata></record>
fulltext fulltext_linktorsrc
identifier DOI: 10.48550/arxiv.2410.08153
ispartof
issn
language eng
recordid cdi_arxiv_primary_2410_08153
source arXiv.org
subjects Mathematics - Group Theory
title Dimension drop in residual chains
url https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2024-12-23T14%3A26%3A17IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-arxiv_GOX&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Dimension%20drop%20in%20residual%20chains&rft.au=Fisher,%20Sam%20P&rft.date=2024-10-10&rft_id=info:doi/10.48550/arxiv.2410.08153&rft_dat=%3Carxiv_GOX%3E2410_08153%3C/arxiv_GOX%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_id=info:pmid/&rfr_iscdi=true