Scale-multiplicative semigroups and geometry: automorphism groups of trees

TL;DR

This paper characterizes the maximal scale-multiplicative semigroups in certain groups acting on trees, linking algebraic properties to geometric features of the trees.

Contribution

It determines the structure of maximal scale-multiplicative semigroups in groups acting 2-transitively on trees' ends, revealing their geometric significance.

Findings

Maximal scale-multiplicative semigroups correspond to geometric features of trees.

The structure of these semigroups is explicitly characterized.

Connections between algebraic semigroup properties and tree geometry are established.

Abstract

A scale-multiplicative semigroup in a totally disconnected, locally compact group $G$ is one for which the restriction of the scale function on $G$ is multiplicative. The maximal scale-multiplicative semigroups in groups acting 2-transitively on the set of ends of trees without leaves are determined in this paper and shown to correspond to geometric features of the tree.