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When loxodromics are pseudo-Anosovs on witnesses

Marissa Chesser

Abstract

In this paper, we prove that for subgroups acting on admissible multiarc and curve graphs and for the handlebody group acting on the disk graph, the loxodromic elements are exactly those for which some pure power is a pseudo-Anosov on a witness. This generalizes the result of Masur and Minsky that the elements of the mapping class group that act loxodromically on the curve graph are the pseudo-Anosov elements.

When loxodromics are pseudo-Anosovs on witnesses

Abstract

In this paper, we prove that for subgroups acting on admissible multiarc and curve graphs and for the handlebody group acting on the disk graph, the loxodromic elements are exactly those for which some pure power is a pseudo-Anosov on a witness. This generalizes the result of Masur and Minsky that the elements of the mapping class group that act loxodromically on the curve graph are the pseudo-Anosov elements.
Paper Structure (10 sections, 19 theorems, 10 equations)

This paper contains 10 sections, 19 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Coarse geometry
  4. Graphs and witnesses
  5. Pseudo-Anosov on a witness implies loxodromic
  6. Distance formulas
  7. The proof that pseudo-Anosov on a witness implies loxodromic
  8. Loxodromic implies pseudo-Anosov on a witness
  9. Admissible multiarc and curve graphs
  10. Disk graphs

Key Result

Theorem 1.1

An element of the mapping class group acts loxodromically on the curve graph if and only if it is a pseudo-Anosov element.

Theorems & Definitions (44)

  • Theorem 1.1: masur-minsky-I
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4: kopreski2023multiarc
  • Definition 2.5
  • ...and 34 more