Scale Groups

TL;DR

This paper explores the relationship between certain closed subgroups of tree isometries, self-replicating groups, and totally disconnected locally compact groups with positive scale, revealing deep structural connections.

Contribution

It establishes a correspondence between specific subgroups fixing an end of a regular tree and self-replicating groups, linking geometric and algebraic group properties.

Findings

Characterization of subgroups fixing an end of the tree

Connection to self-replicating groups acting on rooted trees

Identification of elements with positive scale in locally compact groups

Abstract

Closed subgroups of the group of isometries of the regular tree $\treeq$ that fix an end of the tree and are vertex-transitive are shown to correspond, on one hand, to self-replicating groups acting on rooted trees and, on the other hand, to elements of totally disconnected, locally compact groups having positive scale.