Parabolicity on graphs

Large scale properties of Riemannian manifolds, in particular, those properties preserved by quasi-isometries, can be studied using discrete structures which approximate the manifolds. In a sequence of papers, M. Kanai proved that, under mild conditions, many properties are preserved by a certain (q...

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Detalles Bibliográficos
Autores: Martínez Pérez, Álvaro, Rodríguez, José M.
Tipo de recurso: artículo
Fecha de publicación:2024
País:España
Institución:Universidad de Castilla-La Mancha
Repositorio:RUIdeRA. Repositorio Institucional de la UCLM
OAI Identifier:oai:ruidera.uclm.es:10578/42948
Acceso en línea:https://hdl.handle.net/10578/42948
Access Level:acceso abierto
Palabra clave:Parabolicity
P-parabolicity
Quasi-isometries
Isoperimetric inequality
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spelling Parabolicity on graphsMartínez Pérez, ÁlvaroRodríguez, José M.ParabolicityP-parabolicityQuasi-isometriesIsoperimetric inequalityLarge scale properties of Riemannian manifolds, in particular, those properties preserved by quasi-isometries, can be studied using discrete structures which approximate the manifolds. In a sequence of papers, M. Kanai proved that, under mild conditions, many properties are preserved by a certain (quasi-isometric) graph approximation of a manifold. One of these properties is p-parabolicity. A manifold M (respectively, a graph G) is said to be p-parabolic if all positive p-superharmonic functions on M (resp. G) are constant. This is equivalent to not having p-Green’s function (i.e. a positive fundamental solution of the p-LaplaceBeltrami operator). Herein we study directly the p-parabolicity on graphs. Weobtain some characterizations in terms of graph decompositions. Also, we give necessary and sufficient conditions for a uniform hyperbolic graph to be p-parabolic in terms of its boundary at infinity. Finally, we prove that if a uniform hyperbolic graph satisfies the (Cheeger) isoperimetric inequality, then it is non-p-parabolic for every 1 <p<∞.Springer202520252024info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/10578/42948reponame:RUIdeRA. Repositorio Institucional de la UCLMinstname:Universidad de Castilla-La ManchaInglésPGC2018-098321-B-I00PID2021126124NB-I00PID2019-106433GB-I00AEI / 10.13039/5011 00011033EPUC3M23info:eu-repo/semantics/openAccessAttribution-NonCommercial-NoDerivs 3.0 Spainhttp://creativecommons.org/licenses/by-nc-nd/3.0/es/oai:ruidera.uclm.es:10578/429482026-05-27T07:36:41Z
dc.title.none.fl_str_mv Parabolicity on graphs
title Parabolicity on graphs
spellingShingle Parabolicity on graphs
Martínez Pérez, Álvaro
Parabolicity
P-parabolicity
Quasi-isometries
Isoperimetric inequality
title_short Parabolicity on graphs
title_full Parabolicity on graphs
title_fullStr Parabolicity on graphs
title_full_unstemmed Parabolicity on graphs
title_sort Parabolicity on graphs
dc.creator.none.fl_str_mv Martínez Pérez, Álvaro
Rodríguez, José M.
author Martínez Pérez, Álvaro
author_facet Martínez Pérez, Álvaro
Rodríguez, José M.
author_role author
author2 Rodríguez, José M.
author2_role author
dc.subject.none.fl_str_mv Parabolicity
P-parabolicity
Quasi-isometries
Isoperimetric inequality
topic Parabolicity
P-parabolicity
Quasi-isometries
Isoperimetric inequality
description Large scale properties of Riemannian manifolds, in particular, those properties preserved by quasi-isometries, can be studied using discrete structures which approximate the manifolds. In a sequence of papers, M. Kanai proved that, under mild conditions, many properties are preserved by a certain (quasi-isometric) graph approximation of a manifold. One of these properties is p-parabolicity. A manifold M (respectively, a graph G) is said to be p-parabolic if all positive p-superharmonic functions on M (resp. G) are constant. This is equivalent to not having p-Green’s function (i.e. a positive fundamental solution of the p-LaplaceBeltrami operator). Herein we study directly the p-parabolicity on graphs. Weobtain some characterizations in terms of graph decompositions. Also, we give necessary and sufficient conditions for a uniform hyperbolic graph to be p-parabolic in terms of its boundary at infinity. Finally, we prove that if a uniform hyperbolic graph satisfies the (Cheeger) isoperimetric inequality, then it is non-p-parabolic for every 1 <p<∞.
publishDate 2024
dc.date.none.fl_str_mv 2024
2025
2025
dc.type.none.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/10578/42948
url https://hdl.handle.net/10578/42948
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv PGC2018-098321-B-I00
PID2021126124NB-I00
PID2019-106433GB-I00
AEI / 10.13039/5011 00011033
EPUC3M23
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
Attribution-NonCommercial-NoDerivs 3.0 Spain
http://creativecommons.org/licenses/by-nc-nd/3.0/es/
eu_rights_str_mv openAccess
rights_invalid_str_mv Attribution-NonCommercial-NoDerivs 3.0 Spain
http://creativecommons.org/licenses/by-nc-nd/3.0/es/
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Springer
publisher.none.fl_str_mv Springer
dc.source.none.fl_str_mv reponame:RUIdeRA. Repositorio Institucional de la UCLM
instname:Universidad de Castilla-La Mancha
instname_str Universidad de Castilla-La Mancha
reponame_str RUIdeRA. Repositorio Institucional de la UCLM
collection RUIdeRA. Repositorio Institucional de la UCLM
repository.name.fl_str_mv
repository.mail.fl_str_mv
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