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

General Mathematics

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10.1017/S1446788722000040

Cite this

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AU - Möller, Rögnvaldur G.

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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.

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KW - groups acting on trees

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

Abstract

Keywords

ASJC Scopus subject areas

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