Groups acting on semimetric spaces and quasi-isometries of monoids

Robert Gray; Mark Kambites

doi:10.1090/S0002-9947-2012-05868-5

Groups acting on semimetric spaces and quasi-isometries of monoids

Robert Gray^*, Mark Kambites

^*Corresponding author for this work

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

10 Citations (Scopus)

Abstract

We study groups acting by length-preserving transformations on spaces equipped with asymmetric, partially-defined distance functions. We introduce a natural notion of quasi-isometry for such spaces and exhibit an extension of the Svarc-Milnor lemma to this setting. Among the most natural examples of these spaces are finitely generated monoids and semigroups and their Cayley and Schutzenberger graphs. We apply our results to show that a number of important properties of monoids are quasi-isometry invariants.

Original language	English
Article number	PII S0002-9947(2012)05868-5
Pages (from-to)	555-578
Number of pages	24
Journal	Transactions of the American Mathematical Society
Volume	365
Issue number	2
DOIs	https://doi.org/10.1090/S0002-9947-2012-05868-5
Publication status	Published - Feb 2013

Keywords

Monoid
group
finitely generated
action
semimetric space
quasi-metric space
Des Demi-Groupes
Finiteness conditions
Semigroups
Growth
Ends
Inverse

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10.1090/S0002-9947-2012-05868-5

Cite this

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title = "Groups acting on semimetric spaces and quasi-isometries of monoids",

abstract = "We study groups acting by length-preserving transformations on spaces equipped with asymmetric, partially-defined distance functions. We introduce a natural notion of quasi-isometry for such spaces and exhibit an extension of the Svarc-Milnor lemma to this setting. Among the most natural examples of these spaces are finitely generated monoids and semigroups and their Cayley and Schutzenberger graphs. We apply our results to show that a number of important properties of monoids are quasi-isometry invariants.",

keywords = "Monoid, group, finitely generated, action, semimetric space, quasi-metric space, Des Demi-Groupes, Finiteness conditions, Semigroups, Growth, Ends, Inverse",

author = "Robert Gray and Mark Kambites",

note = "The first author{\textquoteright}s research was supported by an EPSRC Postdoctoral Fellowship. The second author{\textquoteright}s research was supported by an RCUK Academic Fellowship.",

year = "2013",

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doi = "10.1090/S0002-9947-2012-05868-5",

language = "English",

volume = "365",

pages = "555--578",

journal = "Transactions of the American Mathematical Society",

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T1 - Groups acting on semimetric spaces and quasi-isometries of monoids

AU - Gray, Robert

AU - Kambites, Mark

N1 - The first author’s research was supported by an EPSRC Postdoctoral Fellowship. The second author’s research was supported by an RCUK Academic Fellowship.

PY - 2013/2

Y1 - 2013/2

N2 - We study groups acting by length-preserving transformations on spaces equipped with asymmetric, partially-defined distance functions. We introduce a natural notion of quasi-isometry for such spaces and exhibit an extension of the Svarc-Milnor lemma to this setting. Among the most natural examples of these spaces are finitely generated monoids and semigroups and their Cayley and Schutzenberger graphs. We apply our results to show that a number of important properties of monoids are quasi-isometry invariants.

AB - We study groups acting by length-preserving transformations on spaces equipped with asymmetric, partially-defined distance functions. We introduce a natural notion of quasi-isometry for such spaces and exhibit an extension of the Svarc-Milnor lemma to this setting. Among the most natural examples of these spaces are finitely generated monoids and semigroups and their Cayley and Schutzenberger graphs. We apply our results to show that a number of important properties of monoids are quasi-isometry invariants.

KW - Monoid

KW - group

KW - finitely generated

KW - action

KW - semimetric space

KW - quasi-metric space

KW - Des Demi-Groupes

KW - Finiteness conditions

KW - Semigroups

KW - Growth

KW - Ends

KW - Inverse

UR - http://www.ams.org/journals/tran/2013-365-02/S0002-9947-2012-05868-5/

U2 - 10.1090/S0002-9947-2012-05868-5

DO - 10.1090/S0002-9947-2012-05868-5

M3 - Article

SN - 0002-9947

VL - 365

SP - 555

EP - 578

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 2

M1 - PII S0002-9947(2012)05868-5

ER -

Groups acting on semimetric spaces and quasi-isometries of monoids

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Keywords

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Finiteness Conditions and Index: Finiteness Conditions and Index in Semigroups and Monoids

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Groups acting on semimetric spaces and quasi-isometries of monoids

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Keywords

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Finiteness Conditions and Index: Finiteness Conditions and Index in Semigroups and Monoids

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