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 -