Cayley-Abels graphs and invariants of totally disconnected, locally compact groups

ÁRNADÓTTIR, ARNBJÖRG SOFFÍA;Lederle, Waltraud;MÖLLER, RÖGNVALDUR G.
(2022) Australian Mathematical Society. Journal — Vol. np, n° np, p. 1-33 (2022)

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  • ÁRNADÓTTIR, ARNBJÖRG SOFFÍAUniv. of Waterloo, Canada
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  • Lederle, WaltraudUCLouvain
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  • MÖLLER, RÖGNVALDUR G.Univ. of Iceland
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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 Td denotes the d-regular tree, then the minimal degree of Aut(Td) is d for all d≥2 .
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ÁRNADÓTTIR, A. S., Lederle, W., & MÖLLER, R. G. (2022). Cayley-Abels graphs and invariants of totally disconnected, locally compact groups. Australian Mathematical Society. Journal, np(np), 1-33. https://doi.org/10.1017/S1446788722000040 (Original work published 2022)