### 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.

Original language | English |
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Pages (from-to) | 221-265 |

Number of pages | 45 |

Journal | Israel Journal of Mathematics |

Volume | 237 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Mar 2020 |

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## Cite this

*Israel Journal of Mathematics*,

*237*(1), 221-265. https://doi.org/10.1007/s11856-020-2005-0