A graph-theoretic description of scale-multiplicative semigroups of automorphisms
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A GRAPH-THEORETIC DESCRIPTION OF SCALE-MULTIPLICATIVE SEMIGROUPS OF AUTOMORPHISMS BY
Cheryl E. Praeger School of Physics, Mathematics and Computing, The University of Western Australia 35 Stirling Highway, Crawley, WA 6009. Australia e-mail: [email protected] AND
Jacqui Ramagge School of Mathematics and Statistics, Sydney University NSW 2006, Australia e-mail: [email protected] AND
George A. Willis School of Mathematical and Physical Sciences, The University of Newcastle Callaghan, NSW 2308, Australia e-mail: [email protected] 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. M¨ oller that U is tidy for x if and only if a certain graph is a regular, rooted tree.
This research was supported by Australian Research Council grants DP150100060
and DP160102323. Received August 11, 2018 and in revised form June 18, 2019
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C. E. PRAEGER, J. RAMAGGE AND G. A. WILLIS
Isr. J. Math.
Contents
1. Introduction . . . . . . . . . . . . . . 2. Instructive graph-theoretic example 3. Background material and notation . 4. Flat groups and their subsemigroups 5. P -graphs . . . . . . . . . . . . . . . 6. Examples . . . . . . . . . . . . . . . 7. Comments and Questions . . . . . . References . . . . . . . . . . . . . . . . .
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1. Introduction The connected component of the identity of a locally compact group is a normal subgroup and the quotient by this subgroup is totally disconnected and locally compact. Hence every locally compact group is an extension of a connected locally compact group by a totally disconnected, locally compact group. By the solution to Hilbert’s fifth problem in 1952 from the combined work in [4, 7], connected locally compact groups can be approximated from above by Lie groups. The quest for the understanding of connected locally compact groups could then apply linear algebraic techniques via Lie algebras. It was not until 1994 [12] that a corresponding insight was available for totally disconnected, locally compact groups. In [12], Willis introduced the notions of scale function and tidy subgroups in the context of a totally disconnected, locally compact group G. The scale of an automorphism of G is analogous to an eigenvalue for a linear operator, with tidy subgroups being compact ope
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