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Groups acting on graphs with polynomial growth

✍ Scribed by Norbert Seifter

Publisher: Elsevier Science
Year: 1991
Tongue: English
Weight: 820 KB
Volume: 89
Category: Article
ISSN: 0012-365X
DOI: 10.1016/0012-365x(91)90120-q

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✦ Synopsis

Seifter, N., Groups acting on graphs with polynomial growth, Discrete Mathematics 89 (1991) 269-280.

In the first part of this paper we consider nilpotent groups G acting with finitely many orbits on infinite connected locally finite graphs X thereby showing that all (Y E G of infinite order are automorphisms of type 2 of X. In the second part we investigate the automorphism groups of connected locally finite transitive graphs X with polynomial growth thereby showing that AUT(X) is countable if and only if it is finitely generated and nilpotent-by-finite. In this case we also prove that X is contractible to a Cayley graph C(G, H) of a nilpotent group G (for some finite generating set H) which has the same growth degree as X. If X is a transitive strip we show that AUT(X) is uncountable if and only if it contains a finitely generated metabelian subgroup with exponential growth.

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