Abstract
It is shown that a flat subgroup, H, of the totally disconnected, locally compact group G decomposes into a finite number of subsemigroups on which the scale function is multiplicative. The image, P, of a multiplicative semigroup in the quotient, H/H(1), of H by its uniscalar subgroup has a unique minimal generating set which determines a natural Cayley graph structure on P. For each compact, open subgroup U of G, a graph is defined and it is shown that if P is multiplicative over U then this graph is a regular, rooted, strongly simple P-graph. This extends to higher rank the result of R. Möller that U is tidy for x if and only if a certain graph is a regular, rooted tree.
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This research was supported by Australian Research Council grants DP150100060 and DP160102323.
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Praeger, C.E., Ramagge, J. & Willis, G.A. A graph-theoretic description of scale-multiplicative semigroups of automorphisms. Isr. J. Math. 237, 221–265 (2020). https://doi.org/10.1007/s11856-020-2005-0
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DOI: https://doi.org/10.1007/s11856-020-2005-0