TY - JOUR

T1 - A graph-theoretic description of scale-multiplicative semigroups of automorphisms

AU - Praeger, Cheryl E.

AU - Ramagge, Jacqui

AU - Willis, George A.

PY - 2020/3/1

Y1 - 2020/3/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85084825312&partnerID=8YFLogxK

U2 - 10.1007/s11856-020-2005-0

DO - 10.1007/s11856-020-2005-0

M3 - Article

AN - SCOPUS:85084825312

VL - 237

SP - 221

EP - 265

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 1

ER -