Stochastic approximation of lamplighter metrics

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F22%3A00363345" target="_blank" >RIV/68407700:21230/22:00363345 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1112/blms.12657" target="_blank" >https://doi.org/10.1112/blms.12657</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1112/blms.12657" target="_blank" >10.1112/blms.12657</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Stochastic approximation of lamplighter metrics

Popis výsledku v původním jazyce

We observe that embeddings into random metrics can be fruitfully used to study the L-1-embeddability of lamplighter graphs or groups, and more generally lamplighter metric spaces. Once this connection has been established, several new upper bound estimates on the L-1-distortion of lamplighter metrics follow from known related estimates about stochastic embeddings into dominating tree-metrics. For instance, every lamplighter metric on an n-point metric space embeds bi-Lipschitzly into L-1 with distortion O(logn). In particular, for every finite group G the lamplighter group H=Z(2) gimel G bi-Lipschitzly embeds into L-1 with distortion O(loglog|H|). In the case where the ground space in the lamplighter construction is a graph with some topological restrictions, better distortion estimates can be achieved. Finally, we discuss how a coarse embedding into L-1 of the lamplighter group over the d-dimensional infinite lattice Z(d) can be constructed from bi-Lipschitz embeddings of the lamplighter graphs over finite d-dimensional grids, and we include a remark on Lipschitz free spaces over finite metric spaces.

Název v anglickém jazyce

Stochastic approximation of lamplighter metrics

Popis výsledku anglicky

We observe that embeddings into random metrics can be fruitfully used to study the L-1-embeddability of lamplighter graphs or groups, and more generally lamplighter metric spaces. Once this connection has been established, several new upper bound estimates on the L-1-distortion of lamplighter metrics follow from known related estimates about stochastic embeddings into dominating tree-metrics. For instance, every lamplighter metric on an n-point metric space embeds bi-Lipschitzly into L-1 with distortion O(logn). In particular, for every finite group G the lamplighter group H=Z(2) gimel G bi-Lipschitzly embeds into L-1 with distortion O(loglog|H|). In the case where the ground space in the lamplighter construction is a graph with some topological restrictions, better distortion estimates can be achieved. Finally, we discuss how a coarse embedding into L-1 of the lamplighter group over the d-dimensional infinite lattice Z(d) can be constructed from bi-Lipschitz embeddings of the lamplighter graphs over finite d-dimensional grids, and we include a remark on Lipschitz free spaces over finite metric spaces.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2022

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Bulletin of the London Mathematical Society

ISSN

0024-6093

e-ISSN

1469-2120

Svazek periodika

54

Číslo periodika v rámci svazku

5

Stát vydavatele periodika

GB - Spojené království Velké Británie a Severního Irska

Počet stran výsledku

23

Strana od-do

1804-1826

Kód UT WoS článku

000787367900001

EID výsledku v databázi Scopus

2-s2.0-85128812683