A graph-theoretic description of scale-multiplicative semigroups of automorphisms

TL;DR

This paper explores the structure of scale-multiplicative semigroups of automorphisms in totally disconnected, locally compact groups, using graph theory to describe their properties and decompositions.

Contribution

It introduces a graph-theoretic framework to analyze scale-multiplicative semigroups and extends existing results to higher rank cases.

Findings

Decomposition of flat subgroups into finitely many scale-multiplicative subsemigroups.

Construction of Cayley graph structures from minimal generating sets.

Identification of regular, rooted, strongly simple P-graphs under certain conditions.

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.