Geometric invariants of groups and property R-infty
Autor(a) principal: | |
---|---|
Data de Publicação: | 2022 |
Tipo de documento: | Tese |
Idioma: | eng |
Título da fonte: | Repositório Institucional da UFSCAR |
Texto Completo: | https://repositorio.ufscar.br/handle/ufscar/15958 |
Resumo: | In this thesis we study property R_\infty for some classes of finitely generated groups by the use of the BNS invariant \Sigma^1 and some other geometric tools. In the combinatorial chapters of the work (4, 5, 6, 10 and 11), we compute \Sigma^1 for the family of Generalized Solvable Baumslag-Solitar groups \Gamma_n and use it to obtain a new proof of R_\infty for them, by using Gonçalves and Kochloukova's paper. Then, we get nice information on finite index subgroups H of any \Gamma_n by finding suitable generators and a presentation, and by computing their \Sigma^1. This gives a new proof of R_\infty for H and for every finite direct product of such groups. We also show that no nilpotent quotients of the groups \Gamma_n have R_\infty. With a help of Cashen and Levitt's paper, we give an algorithmic classification of all possible shapes for \Sigma^1 of GBS and GBS_n groups and show how to use it to obtain some partial twisted-conjugacy information in some specific cases. Furthermore, we show that the existence of certain spherically convex and invariant k-dimensional polytopes in the character sphere of a finitely generated group G can guarantee R_\infty for G. In the geometric chapters (7 through 9), we study property R_\infty for hyperbolic and relatively hyperbolic groups. First, we give a didactic presentation of the (already known) proof of R_\infty for hyperbolic groups given by Levitt and Lustig (which also uses a paper from Paulin). Then, we expand and analyse the sketch of proof of R_\infty for relatively hyperbolic groups given by A. Fel'shtyn on his survey paper: we point out the valid arguments and difficulties of the proof, exhibit what would be a complete proof based on his sketch and show an example where the proof method doesn't work. |
id |
SCAR_221d155e27f2d86b466101455948280c |
---|---|
oai_identifier_str |
oai:repositorio.ufscar.br:ufscar/15958 |
network_acronym_str |
SCAR |
network_name_str |
Repositório Institucional da UFSCAR |
repository_id_str |
4322 |
spelling |
Sgobbi, Wagner CarvalhoVendrúscolo, Danielhttp://lattes.cnpq.br/8602232587914830Wong, Peter Ngai-Singhttp://lattes.cnpq.br/9104201992938700http://lattes.cnpq.br/8536818102991005f433958b-5589-48d4-a4b6-45c44b2450662022-05-02T15:43:49Z2022-05-02T15:43:49Z2022-01-05SGOBBI, Wagner Carvalho. Geometric invariants of groups and property R-infty. 2022. Tese (Doutorado em Matemática) – Universidade Federal de São Carlos, São Carlos, 2022. Disponível em: https://repositorio.ufscar.br/handle/ufscar/15958.https://repositorio.ufscar.br/handle/ufscar/15958In this thesis we study property R_\infty for some classes of finitely generated groups by the use of the BNS invariant \Sigma^1 and some other geometric tools. In the combinatorial chapters of the work (4, 5, 6, 10 and 11), we compute \Sigma^1 for the family of Generalized Solvable Baumslag-Solitar groups \Gamma_n and use it to obtain a new proof of R_\infty for them, by using Gonçalves and Kochloukova's paper. Then, we get nice information on finite index subgroups H of any \Gamma_n by finding suitable generators and a presentation, and by computing their \Sigma^1. This gives a new proof of R_\infty for H and for every finite direct product of such groups. We also show that no nilpotent quotients of the groups \Gamma_n have R_\infty. With a help of Cashen and Levitt's paper, we give an algorithmic classification of all possible shapes for \Sigma^1 of GBS and GBS_n groups and show how to use it to obtain some partial twisted-conjugacy information in some specific cases. Furthermore, we show that the existence of certain spherically convex and invariant k-dimensional polytopes in the character sphere of a finitely generated group G can guarantee R_\infty for G. In the geometric chapters (7 through 9), we study property R_\infty for hyperbolic and relatively hyperbolic groups. First, we give a didactic presentation of the (already known) proof of R_\infty for hyperbolic groups given by Levitt and Lustig (which also uses a paper from Paulin). Then, we expand and analyse the sketch of proof of R_\infty for relatively hyperbolic groups given by A. Fel'shtyn on his survey paper: we point out the valid arguments and difficulties of the proof, exhibit what would be a complete proof based on his sketch and show an example where the proof method doesn't work.Nesta tese estudamos a propriedade R_\infty para algumas classes de grupos finitamente gerados através do uso do BNS invariante \Sigma^1 e de algumas outras ferramentas geométricas. Nos capítulos combinatórios do trabalho (4, 5, 6, 10 e 11), computamos \Sigma^1 para a família dos grupos de Baumslag-Solitar solúveis generalizados \Gamma_n e o usamos para obter uma nova prova de R_\infty para tais grupos, usando o artigo de Gonçalves e Kochloukova. Então, obtemos boas informações sobre os subgrupos H de índice finito de qualquer \Gamma_n encontrando geradores adequados, uma presentação e computando seu \Sigma^1. Com isto, obtemos uma nova prova de R_\infty para H e para qualquer produto direto finito de tais grupos. Também provamos que nenhum quociente nilpotente dos grupos \Gamma_n tem R_\infty. Com a ajuda do artigo de Cashen e Levitt, damos uma classificação algorítmica de todos os possíveis formatos do invariante \Sigma^1 para grupos GBS e GBS_n e mostramos como usá-lo para obter algumas informações parciais sobre classes de conjugação torcida em alguns casos específicos. Além disso, provamos que a existência de certos poliedros esfericamente convexos e invariantes na esfera de caracteres de um grupo finitamente gerado arbitrário G pode garantir R_\infty para G. Nos capítulos geométricos (7 a 9), estudamos a propriedade R_\infty para grupos hiperbólicos e relativamente hiperbólicos. Primeiro, apresentamos de forma didática a prova (já conhecida) de R_\infty para grupos hiperbólicos dada por Levitt e Lustig (que também usa um artigo de Paulin). Então, expandimos e analisamos o rascunho de prova de R_\infty para grupos relativamente hiperbólicos dado por Fel'shtyn em seu artigo: mostramos os argumentos válidos e as dificuldades da prova, exibimos como seria uma prova completa baseada em seu rascunho e damos um exemplo onde tal método de prova não funciona.Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Processo n° 2017/21208-0, Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Processo BEPE n° 2019/03150-0, Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)CAPES: Código de Financiamento 001engUniversidade Federal de São CarlosCâmpus São CarlosPrograma de Pós-Graduação em Matemática - PPGMUFSCarAttribution-NonCommercial-NoDerivs 3.0 Brazilhttp://creativecommons.org/licenses/by-nc-nd/3.0/br/info:eu-repo/semantics/openAccessPropriedade R-infinitoTopologiaInvariantes BNSTeoria combinatória de gruposTeoria geométrica de gruposProperty R-inftyTopologyBNS invariantsCombinatorial group theoryGeometric group theoryCIENCIAS EXATAS E DA TERRA::MATEMATICA::GEOMETRIA E TOPOLOGIAGeometric invariants of groups and property R-inftyInvariantes geométricos de grupos e a propriedade R-infinitoinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesis600600ed7bc463-53db-4665-b793-bc87c9876244reponame:Repositório Institucional da UFSCARinstname:Universidade Federal de São Carlos (UFSCAR)instacron:UFSCARORIGINALWagner_Carvalho_Sgobbi_tese_corrigida_2.pdfWagner_Carvalho_Sgobbi_tese_corrigida_2.pdfVersão final da tese (corrigida)application/pdf4192062https://repositorio.ufscar.br/bitstream/ufscar/15958/3/Wagner_Carvalho_Sgobbi_tese_corrigida_2.pdfeba6975a04e01dd87fbb0ff8388c7c7cMD53modelo_carta-comprovantelogodosppgs-3.pdfmodelo_carta-comprovantelogodosppgs-3.pdfCarta comprovanteapplication/pdf91757https://repositorio.ufscar.br/bitstream/ufscar/15958/6/modelo_carta-comprovantelogodosppgs-3.pdfbdc4c999ce9c6eaad675828dce179460MD56CC-LICENSElicense_rdflicense_rdfapplication/rdf+xml; charset=utf-8811https://repositorio.ufscar.br/bitstream/ufscar/15958/7/license_rdfe39d27027a6cc9cb039ad269a5db8e34MD57TEXTWagner_Carvalho_Sgobbi_tese_corrigida_2.pdf.txtWagner_Carvalho_Sgobbi_tese_corrigida_2.pdf.txtExtracted texttext/plain589052https://repositorio.ufscar.br/bitstream/ufscar/15958/8/Wagner_Carvalho_Sgobbi_tese_corrigida_2.pdf.txt4ccaa954782df842cbf154677f82e06dMD58modelo_carta-comprovantelogodosppgs-3.pdf.txtmodelo_carta-comprovantelogodosppgs-3.pdf.txtExtracted texttext/plain1359https://repositorio.ufscar.br/bitstream/ufscar/15958/10/modelo_carta-comprovantelogodosppgs-3.pdf.txt7242e786af9f92dc58a83807bda28f17MD510THUMBNAILWagner_Carvalho_Sgobbi_tese_corrigida_2.pdf.jpgWagner_Carvalho_Sgobbi_tese_corrigida_2.pdf.jpgIM Thumbnailimage/jpeg6842https://repositorio.ufscar.br/bitstream/ufscar/15958/9/Wagner_Carvalho_Sgobbi_tese_corrigida_2.pdf.jpg810cb629606086a68bbaec703632b7d0MD59modelo_carta-comprovantelogodosppgs-3.pdf.jpgmodelo_carta-comprovantelogodosppgs-3.pdf.jpgIM Thumbnailimage/jpeg5955https://repositorio.ufscar.br/bitstream/ufscar/15958/11/modelo_carta-comprovantelogodosppgs-3.pdf.jpge30bcd75e80683c62ce1ab366a3ef542MD511ufscar/159582023-09-18 18:32:27.941oai:repositorio.ufscar.br:ufscar/15958Repositório InstitucionalPUBhttps://repositorio.ufscar.br/oai/requestopendoar:43222023-09-18T18:32:27Repositório Institucional da UFSCAR - Universidade Federal de São Carlos (UFSCAR)false |
dc.title.eng.fl_str_mv |
Geometric invariants of groups and property R-infty |
dc.title.alternative.por.fl_str_mv |
Invariantes geométricos de grupos e a propriedade R-infinito |
title |
Geometric invariants of groups and property R-infty |
spellingShingle |
Geometric invariants of groups and property R-infty Sgobbi, Wagner Carvalho Propriedade R-infinito Topologia Invariantes BNS Teoria combinatória de grupos Teoria geométrica de grupos Property R-infty Topology BNS invariants Combinatorial group theory Geometric group theory CIENCIAS EXATAS E DA TERRA::MATEMATICA::GEOMETRIA E TOPOLOGIA |
title_short |
Geometric invariants of groups and property R-infty |
title_full |
Geometric invariants of groups and property R-infty |
title_fullStr |
Geometric invariants of groups and property R-infty |
title_full_unstemmed |
Geometric invariants of groups and property R-infty |
title_sort |
Geometric invariants of groups and property R-infty |
author |
Sgobbi, Wagner Carvalho |
author_facet |
Sgobbi, Wagner Carvalho |
author_role |
author |
dc.contributor.authorlattes.por.fl_str_mv |
http://lattes.cnpq.br/8536818102991005 |
dc.contributor.author.fl_str_mv |
Sgobbi, Wagner Carvalho |
dc.contributor.advisor1.fl_str_mv |
Vendrúscolo, Daniel |
dc.contributor.advisor1Lattes.fl_str_mv |
http://lattes.cnpq.br/8602232587914830 |
dc.contributor.advisor-co1.fl_str_mv |
Wong, Peter Ngai-Sing |
dc.contributor.advisor-co1Lattes.fl_str_mv |
http://lattes.cnpq.br/9104201992938700 |
dc.contributor.authorID.fl_str_mv |
f433958b-5589-48d4-a4b6-45c44b245066 |
contributor_str_mv |
Vendrúscolo, Daniel Wong, Peter Ngai-Sing |
dc.subject.por.fl_str_mv |
Propriedade R-infinito Topologia Invariantes BNS Teoria combinatória de grupos Teoria geométrica de grupos |
topic |
Propriedade R-infinito Topologia Invariantes BNS Teoria combinatória de grupos Teoria geométrica de grupos Property R-infty Topology BNS invariants Combinatorial group theory Geometric group theory CIENCIAS EXATAS E DA TERRA::MATEMATICA::GEOMETRIA E TOPOLOGIA |
dc.subject.eng.fl_str_mv |
Property R-infty Topology BNS invariants Combinatorial group theory Geometric group theory |
dc.subject.cnpq.fl_str_mv |
CIENCIAS EXATAS E DA TERRA::MATEMATICA::GEOMETRIA E TOPOLOGIA |
description |
In this thesis we study property R_\infty for some classes of finitely generated groups by the use of the BNS invariant \Sigma^1 and some other geometric tools. In the combinatorial chapters of the work (4, 5, 6, 10 and 11), we compute \Sigma^1 for the family of Generalized Solvable Baumslag-Solitar groups \Gamma_n and use it to obtain a new proof of R_\infty for them, by using Gonçalves and Kochloukova's paper. Then, we get nice information on finite index subgroups H of any \Gamma_n by finding suitable generators and a presentation, and by computing their \Sigma^1. This gives a new proof of R_\infty for H and for every finite direct product of such groups. We also show that no nilpotent quotients of the groups \Gamma_n have R_\infty. With a help of Cashen and Levitt's paper, we give an algorithmic classification of all possible shapes for \Sigma^1 of GBS and GBS_n groups and show how to use it to obtain some partial twisted-conjugacy information in some specific cases. Furthermore, we show that the existence of certain spherically convex and invariant k-dimensional polytopes in the character sphere of a finitely generated group G can guarantee R_\infty for G. In the geometric chapters (7 through 9), we study property R_\infty for hyperbolic and relatively hyperbolic groups. First, we give a didactic presentation of the (already known) proof of R_\infty for hyperbolic groups given by Levitt and Lustig (which also uses a paper from Paulin). Then, we expand and analyse the sketch of proof of R_\infty for relatively hyperbolic groups given by A. Fel'shtyn on his survey paper: we point out the valid arguments and difficulties of the proof, exhibit what would be a complete proof based on his sketch and show an example where the proof method doesn't work. |
publishDate |
2022 |
dc.date.accessioned.fl_str_mv |
2022-05-02T15:43:49Z |
dc.date.available.fl_str_mv |
2022-05-02T15:43:49Z |
dc.date.issued.fl_str_mv |
2022-01-05 |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/doctoralThesis |
format |
doctoralThesis |
status_str |
publishedVersion |
dc.identifier.citation.fl_str_mv |
SGOBBI, Wagner Carvalho. Geometric invariants of groups and property R-infty. 2022. Tese (Doutorado em Matemática) – Universidade Federal de São Carlos, São Carlos, 2022. Disponível em: https://repositorio.ufscar.br/handle/ufscar/15958. |
dc.identifier.uri.fl_str_mv |
https://repositorio.ufscar.br/handle/ufscar/15958 |
identifier_str_mv |
SGOBBI, Wagner Carvalho. Geometric invariants of groups and property R-infty. 2022. Tese (Doutorado em Matemática) – Universidade Federal de São Carlos, São Carlos, 2022. Disponível em: https://repositorio.ufscar.br/handle/ufscar/15958. |
url |
https://repositorio.ufscar.br/handle/ufscar/15958 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.confidence.fl_str_mv |
600 600 |
dc.relation.authority.fl_str_mv |
ed7bc463-53db-4665-b793-bc87c9876244 |
dc.rights.driver.fl_str_mv |
Attribution-NonCommercial-NoDerivs 3.0 Brazil http://creativecommons.org/licenses/by-nc-nd/3.0/br/ info:eu-repo/semantics/openAccess |
rights_invalid_str_mv |
Attribution-NonCommercial-NoDerivs 3.0 Brazil http://creativecommons.org/licenses/by-nc-nd/3.0/br/ |
eu_rights_str_mv |
openAccess |
dc.publisher.none.fl_str_mv |
Universidade Federal de São Carlos Câmpus São Carlos |
dc.publisher.program.fl_str_mv |
Programa de Pós-Graduação em Matemática - PPGM |
dc.publisher.initials.fl_str_mv |
UFSCar |
publisher.none.fl_str_mv |
Universidade Federal de São Carlos Câmpus São Carlos |
dc.source.none.fl_str_mv |
reponame:Repositório Institucional da UFSCAR instname:Universidade Federal de São Carlos (UFSCAR) instacron:UFSCAR |
instname_str |
Universidade Federal de São Carlos (UFSCAR) |
instacron_str |
UFSCAR |
institution |
UFSCAR |
reponame_str |
Repositório Institucional da UFSCAR |
collection |
Repositório Institucional da UFSCAR |
bitstream.url.fl_str_mv |
https://repositorio.ufscar.br/bitstream/ufscar/15958/3/Wagner_Carvalho_Sgobbi_tese_corrigida_2.pdf https://repositorio.ufscar.br/bitstream/ufscar/15958/6/modelo_carta-comprovantelogodosppgs-3.pdf https://repositorio.ufscar.br/bitstream/ufscar/15958/7/license_rdf https://repositorio.ufscar.br/bitstream/ufscar/15958/8/Wagner_Carvalho_Sgobbi_tese_corrigida_2.pdf.txt https://repositorio.ufscar.br/bitstream/ufscar/15958/10/modelo_carta-comprovantelogodosppgs-3.pdf.txt https://repositorio.ufscar.br/bitstream/ufscar/15958/9/Wagner_Carvalho_Sgobbi_tese_corrigida_2.pdf.jpg https://repositorio.ufscar.br/bitstream/ufscar/15958/11/modelo_carta-comprovantelogodosppgs-3.pdf.jpg |
bitstream.checksum.fl_str_mv |
eba6975a04e01dd87fbb0ff8388c7c7c bdc4c999ce9c6eaad675828dce179460 e39d27027a6cc9cb039ad269a5db8e34 4ccaa954782df842cbf154677f82e06d 7242e786af9f92dc58a83807bda28f17 810cb629606086a68bbaec703632b7d0 e30bcd75e80683c62ce1ab366a3ef542 |
bitstream.checksumAlgorithm.fl_str_mv |
MD5 MD5 MD5 MD5 MD5 MD5 MD5 |
repository.name.fl_str_mv |
Repositório Institucional da UFSCAR - Universidade Federal de São Carlos (UFSCAR) |
repository.mail.fl_str_mv |
|
_version_ |
1802136402779963392 |