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Out($F_r$) train track automata I: Proper full fold decompositions

Catherine Eva Pfaff

TL;DR

This work develops a comprehensive framework of train track automata for fully irreducible outer automorphisms of $Out(F_r)$, with a focus on lone-axis cases and proper full fold (pff) decompositions of fully singular train track representatives. It introduces lamination train track (ltt) structures and ideal Whitehead graphs as invariants to organize and encode the dynamics, while linking Stallings fold decompositions to geodesics in Outer space $CV_r$. The authors construct two principal families of automata: lone axis automata $\,\mathcal{A}(IW(\varphi))$ and fully singular pff ltt automata, showing that loops correspond to train track maps representing ageometric fully irreducible outer automorphisms and yield geodesics in $CV_r$. The framework provides a stratified, combinatorial approach to studying axis structure, vertex merging avoidance, and lamination behavior, with potential for broad applications in understanding dynamics of $Out(F_r)$ on Outer space.

Abstract

We describe train track automata for large classes of fully irreducible elements of Out($F_r$), and their associated geodesics in Culler-Vogtmann Outer Space.

Out($F_r$) train track automata I: Proper full fold decompositions

TL;DR

This work develops a comprehensive framework of train track automata for fully irreducible outer automorphisms of , with a focus on lone-axis cases and proper full fold (pff) decompositions of fully singular train track representatives. It introduces lamination train track (ltt) structures and ideal Whitehead graphs as invariants to organize and encode the dynamics, while linking Stallings fold decompositions to geodesics in Outer space . The authors construct two principal families of automata: lone axis automata and fully singular pff ltt automata, showing that loops correspond to train track maps representing ageometric fully irreducible outer automorphisms and yield geodesics in . The framework provides a stratified, combinatorial approach to studying axis structure, vertex merging avoidance, and lamination behavior, with potential for broad applications in understanding dynamics of on Outer space.

Abstract

We describe train track automata for large classes of fully irreducible elements of Out(), and their associated geodesics in Culler-Vogtmann Outer Space.
Paper Structure (60 sections, 27 theorems, 40 equations, 4 tables)

This paper contains 60 sections, 27 theorems, 40 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Edge Maps on Graphs
  4. Directions, turns, & edge paths
  5. Graph maps
  6. Taken turns & tight maps
  7. Directions maps & gate structures
  8. Markings & representatives
  9. Train track (tt) maps and fully irreducible outer automorphisms
  10. Train track (tt) maps
  11. Transition matrices
  12. (Fully) irreducible outer automorphisms
  13. (Periodic) Nielsen paths
  14. Rotationless powers
  15. PNP detection
  16. ...and 45 more sections

Key Result

Lemma 2.1

Let $g \colon \Gamma\to\Gamma$ be an expanding irreducible train track map and $\rho$ an iNP for $g$. Then $\rho=\bar{\rho}_1\rho_2$, where $\rho_1$ and $\rho_2$ are nontrivial legal paths originating at a common vertex $v$ and such that the turn at $v$ between $\rho_1$ and $\rho_2$ is a nondegenera

Theorems & Definitions (53)

  • Lemma 2.1: bh92
  • Lemma 2.2: stablestrata
  • Lemma 2.3
  • proof
  • Proposition 3.1: IWGII, stablestrata
  • Lemma 3.2: Index deficit values
  • proof
  • Proposition 4.1
  • Proposition 4.2: loneaxes
  • Lemma 5.1
  • ...and 43 more