Stability of homomorphisms, coverings and cocycles I: equivalence

Michael Chapman; Alexander Lubotzky

doi:10.1007/s40879-024-00803-8

Stability of homomorphisms, coverings and cocycles I: equivalence

Michael Chapman^*, Alexander Lubotzky

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

This paper is motivated by recent developments in group stability, high dimensional expansion, local testability of error correcting codes and topological property testing. In Part I, we formulate and motivate three stability problems: Homomorphism stability: Are almost homomorphisms close to homomorphisms? Covering stability: Are almost coverings of a cell complex close to genuine coverings of it? Cocycle stability: Are 1-cochains whose coboundary is small close to 1-cocycles? We then prove that these three problems are equivalent.

Original language	English
Article number	13
Journal	European Journal of Mathematics
Volume	11
Issue number	1
DOIs	https://doi.org/10.1007/s40879-024-00803-8
State	Published - Mar 2025
Externally published	Yes

Bibliographical note

Publisher Copyright:
© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2025.

Keywords

Covering theory
Geometric group theory
Group cohomology with non-commutative coefficients
Group stability

Access to Document

10.1007/s40879-024-00803-8

Cite this

@article{832c59fe268846c69195bb5ae9c23bad,

title = "Stability of homomorphisms, coverings and cocycles I: equivalence",

abstract = "This paper is motivated by recent developments in group stability, high dimensional expansion, local testability of error correcting codes and topological property testing. In Part I, we formulate and motivate three stability problems: Homomorphism stability: Are almost homomorphisms close to homomorphisms? Covering stability: Are almost coverings of a cell complex close to genuine coverings of it? Cocycle stability: Are 1-cochains whose coboundary is small close to 1-cocycles? We then prove that these three problems are equivalent.",

keywords = "Covering theory, Geometric group theory, Group cohomology with non-commutative coefficients, Group stability",

author = "Michael Chapman and Alexander Lubotzky",

note = "Publisher Copyright: {\textcopyright} The Author(s), under exclusive licence to Springer Nature Switzerland AG 2025.",

year = "2025",

month = mar,

doi = "10.1007/s40879-024-00803-8",

language = "אנגלית",

volume = "11",

journal = "European Journal of Mathematics",

issn = "2199-675X",

publisher = "Springer International Publishing AG",

number = "1",

}

TY - JOUR

T1 - Stability of homomorphisms, coverings and cocycles I

T2 - equivalence

AU - Chapman, Michael

AU - Lubotzky, Alexander

N1 - Publisher Copyright: © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2025.

PY - 2025/3

Y1 - 2025/3

N2 - This paper is motivated by recent developments in group stability, high dimensional expansion, local testability of error correcting codes and topological property testing. In Part I, we formulate and motivate three stability problems: Homomorphism stability: Are almost homomorphisms close to homomorphisms? Covering stability: Are almost coverings of a cell complex close to genuine coverings of it? Cocycle stability: Are 1-cochains whose coboundary is small close to 1-cocycles? We then prove that these three problems are equivalent.

AB - This paper is motivated by recent developments in group stability, high dimensional expansion, local testability of error correcting codes and topological property testing. In Part I, we formulate and motivate three stability problems: Homomorphism stability: Are almost homomorphisms close to homomorphisms? Covering stability: Are almost coverings of a cell complex close to genuine coverings of it? Cocycle stability: Are 1-cochains whose coboundary is small close to 1-cocycles? We then prove that these three problems are equivalent.

KW - Covering theory

KW - Geometric group theory

KW - Group cohomology with non-commutative coefficients

KW - Group stability

UR - http://www.scopus.com/inward/record.url?scp=85218245149&partnerID=8YFLogxK

U2 - 10.1007/s40879-024-00803-8

DO - 10.1007/s40879-024-00803-8

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:85218245149

SN - 2199-675X

VL - 11

JO - European Journal of Mathematics

JF - European Journal of Mathematics

IS - 1

M1 - 13

ER -

Stability of homomorphisms, coverings and cocycles I: equivalence

Abstract

Bibliographical note

Keywords

Access to Document

Other files and links

Fingerprint

Cite this