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Edge Inversions in $(P_k)$-closed Groups

Kirwin Hampshire, Florian Lehner, Andrew Wood

Abstract

We construct $(P_2)$-closed groups acting on $T_3$ in which all edge inversions have infinite order. This provides a negative answer to a question posed by Tornier. We also construct a family of $(P_2)$-closed groups for which the smallest order of an edge inversion is an arbitrarily high finite number.

Edge Inversions in $(P_k)$-closed Groups

Abstract

We construct -closed groups acting on in which all edge inversions have infinite order. This provides a negative answer to a question posed by Tornier. We also construct a family of -closed groups for which the smallest order of an edge inversion is an arbitrarily high finite number.
Paper Structure (6 sections, 8 theorems, 3 equations, 2 figures)

This paper contains 6 sections, 8 theorems, 3 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Results
  4. A $(P_2)$-closed Group With No Finite Order Inversions
  5. Inversion Orders in $(P_2)$-closures
  6. A Vertex-transitive $(P_2)$-closed Group With No Finite Order Inversions

Key Result

Proposition 3.1

If a $(P)$-closed group $G$ acting on a tree $T$ contains an edge inversion, then $G$ contains an edge inversion of order 2.

Figures (2)

  • Figure 1: The gadget with which we replace each vertex in $T_4$.
  • Figure 2: A gadget and its arcs to adjacent gadgets in $T_4^+$.

Theorems & Definitions (14)

  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • proof
  • Corollary 3.4
  • proof
  • Theorem 3.5
  • proof
  • ...and 4 more