Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Scale Groups

George A. Willis

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.

Scale Groups

Abstract

Closed subgroups of the group of isometries of the regular tree 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.

Paper Structure

This paper contains 18 sections, 20 theorems, 61 equations, 1 table.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Scale groups and self-replicating groups
  4. Scale groups
  5. Self-replicating groups
  6. Note:
  7. The regular tree in terms of sequences of labels
  8. Edge labellings and scale groups, correspondences with self-replicating groups
  9. Edge labellings compatible with a scale group
  10. $P$-labellings and self-replicating groups
  11. Residue actions of a scale group
  12. The tree representation theorem
  13. Construction of the tree and compatible labellings
  14. Uniqueness of scale group representations
  15. Examples of tree representation
  16. ...and 3 more sections

Key Result

Proposition 2.2

Let $G$ be a closed subgroup of $\mathop{\mathrm{Isom}}\nolimits(T_{q+1})$ that fixes the end $\omega$. Then $G^{(e)}$ is a closed subgroup of $G$ and either $G = G^{(e)}$ or there is a translation $x\in G$ with $\omega$ as a repelling end such that $G = G^{(e)}\rtimes \langle x\rangle$. $\Box$

Theorems & Definitions (39)

  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Corollary 2.5
  • Proposition 2.6
  • Example 2.7
  • Definition 2.10
  • Definition 2.12
  • Definition 2.13
  • ...and 29 more