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

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_ddenotes the d-regular tree, then the minimal degree of Aut(T_d) is d for all d ≥ 2.

Original language	English
Pages (from-to)	145-177
Number of pages	33
Journal	Journal of the Australian Mathematical Society
Volume	114
Issue number	2
DOIs	https://doi.org/10.1017/S1446788722000040
State	Published - 13 Apr 2023
Externally published	Yes

Keywords

Cayley-Abels graphs
groups acting on trees
modular function
scale function
totally disconnected locally compact groups

ASJC Scopus subject areas

Mathematics (all)

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 Tddenotes the d-regular tree, then the minimal degree of Aut(Td) is d for all d ≥ 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. 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} 2023 Cambridge University Press. All rights reserved.",

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doi = "10.1017/S1446788722000040",

language = "English",

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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. 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: © 2023 Cambridge University Press. All rights reserved.

PY - 2023/4/13

Y1 - 2023/4/13

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 Tddenotes the d-regular tree, then the minimal degree of Aut(Td) is d for all d ≥ 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 Tddenotes the d-regular tree, then the minimal degree of Aut(Td) is d for all d ≥ 2.

KW - Cayley-Abels graphs

KW - groups acting on trees

KW - modular function

KW - scale function

KW - totally disconnected locally compact groups

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CAYLEY-ABELS GRAPHS AND INVARIANTS OF TOTALLY DISCONNECTED, LOCALLY COMPACT GROUPS

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Keywords

ASJC Scopus subject areas

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