CAYLEY-ABELS GRAPHS and INVARIANTS of TOTALLY DISCONNECTED, LOCALLY COMPACT GROUPS

Arnbjörg Soffía Árnadóttir; Waltraud Lederle; Rögnvaldur G. Möller

doi:10.1017/S1446788722000040

CAYLEY-ABELS GRAPHS and INVARIANTS of TOTALLY DISCONNECTED, LOCALLY COMPACT GROUPS

Arnbjörg Soffía Árnadóttir, Waltraud Lederle, Rögnvaldur G. Möller

Faculty of Physical Sciences

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

Abstract

A connected, locally finite graph Γ is a Cayley-Abels graph for a totally disconnected, locally compact group G if G acts vertex-transitively on Γ with compact, open vertex stabilizers. Define the minimal degree of G as the minimal degree of a Cayley-Abels graph of G. We relate the minimal degree in various ways to the modular function, the scale function and the structure of compact open subgroups. As an application, we prove that if T_(d) denotes the d-regular tree, then the minimal degree of T_(d) is d for all d\geq 2.

Original language	English
Journal	Journal of the Australian Mathematical Society
DOIs	https://doi.org/10.1017/S1446788722000040
Publication status	Accepted/In press - 2022

Bibliographical note

Funding Information:
The second named author was supported by Early Postdoc. Mobility scholarship No. 175106 from the Swiss National Science Foundation. Part of this work was done when she was visiting the University of Newcastle with the International Visitor Program of the Sydney Mathematical Research Institute.

Publisher Copyright:
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Other keywords

05C25
2020 Mathematics subject classification
20E08
22D05
22F50

Access to Document

10.1017/S1446788722000040

Cite this

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title = "CAYLEY-ABELS GRAPHS and INVARIANTS of TOTALLY DISCONNECTED, LOCALLY COMPACT GROUPS",

abstract = "A connected, locally finite graph Γ is a Cayley-Abels graph for a totally disconnected, locally compact group G if G acts vertex-transitively on Γ with compact, open vertex stabilizers. Define the minimal degree of G as the minimal degree of a Cayley-Abels graph of G. We relate the minimal degree in various ways to the modular function, the scale function and the structure of compact open subgroups. As an application, we prove that if T_(d) denotes the d-regular tree, then the minimal degree of T_(d) is d for all d\geq 2.",

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note = "Funding Information: The second named author was supported by Early Postdoc. Mobility scholarship No. 175106 from the Swiss National Science Foundation. Part of this work was done when she was visiting the University of Newcastle with the International Visitor Program of the Sydney Mathematical Research Institute. Publisher Copyright: {\textcopyright} The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.",

year = "2022",

doi = "10.1017/S1446788722000040",

language = "English",

journal = "Journal of the Australian Mathematical Society",

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AU - Árnadóttir, Arnbjörg Soffía

AU - Lederle, Waltraud

AU - Möller, Rögnvaldur G.

N1 - Funding Information: The second named author was supported by Early Postdoc. Mobility scholarship No. 175106 from the Swiss National Science Foundation. Part of this work was done when she was visiting the University of Newcastle with the International Visitor Program of the Sydney Mathematical Research Institute. Publisher Copyright: © The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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N2 - A connected, locally finite graph Γ is a Cayley-Abels graph for a totally disconnected, locally compact group G if G acts vertex-transitively on Γ with compact, open vertex stabilizers. Define the minimal degree of G as the minimal degree of a Cayley-Abels graph of G. We relate the minimal degree in various ways to the modular function, the scale function and the structure of compact open subgroups. As an application, we prove that if T_(d) denotes the d-regular tree, then the minimal degree of T_(d) is d for all d\geq 2.

AB - A connected, locally finite graph Γ is a Cayley-Abels graph for a totally disconnected, locally compact group G if G acts vertex-transitively on Γ with compact, open vertex stabilizers. Define the minimal degree of G as the minimal degree of a Cayley-Abels graph of G. We relate the minimal degree in various ways to the modular function, the scale function and the structure of compact open subgroups. As an application, we prove that if T_(d) denotes the d-regular tree, then the minimal degree of T_(d) is d for all d\geq 2.

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KW - 2020 Mathematics subject classification

KW - 20E08

KW - 22D05

KW - 22F50

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M3 - Article

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JF - Journal of the Australian Mathematical Society

ER -

CAYLEY-ABELS GRAPHS and INVARIANTS of TOTALLY DISCONNECTED, LOCALLY COMPACT GROUPS

Abstract

Bibliographical note

Other keywords

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