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Certifying Anosov representations

J. Maxwell Riestenberg

Abstract

By providing new finite criteria which certify that a finitely generated subgroup of $\mathrm{SL}(d,\mathbb{R})$ or $\mathrm{SL}(d,\mathbb{C})$ is projective Anosov, we obtain a practical algorithm to verify the Anosov condition. We demonstrate on a surface group of genus 2 in $\mathrm{SL}(3,\mathbb{R})$ by verifying the criteria for all words of length 8. The previous version required checking all words of length $2$ million.

Certifying Anosov representations

Abstract

By providing new finite criteria which certify that a finitely generated subgroup of or is projective Anosov, we obtain a practical algorithm to verify the Anosov condition. We demonstrate on a surface group of genus 2 in by verifying the criteria for all words of length 8. The previous version required checking all words of length million.
Paper Structure (17 sections, 6 theorems, 64 equations, 1 figure)

This paper contains 17 sections, 6 theorems, 64 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Symmetric space reminders
  4. The model and metric
  5. Geodesics and vector-valued distance
  6. Scale of the metric
  7. The visual boundary
  8. Projective space
  9. Regular directions and root pseudometric
  10. Busemann functions centered at line-hyperplane pairs
  11. Transversality and parallel sets
  12. Angles and distances to parallel sets
  13. Angle-to-distance formula
  14. From rays to parallel sets
  15. Controlling zeta-angles
  16. ...and 2 more sections

Key Result

Lemma 3.2

Let $\tau \in \mathbb{KP}^{d-1}$. If $y \in V(x,\mathop{\mathrm{st}}\nolimits(\tau))$, then

Figures (1)

  • Figure 1: Points $w,x,y,z$ in a maximal flat and corresponding directions of types $\zeta$ and $\iota \zeta$.

Theorems & Definitions (18)

  • Lemma 3.2: Root pseudometric and Busemann functions
  • proof
  • Example 3.3
  • Lemma 4.1: Angle-to-distance formula
  • proof
  • Remark 4.2
  • Lemma 4.3: Detecting transversality
  • proof
  • Lemma 4.4: Distance from ray to parallel set
  • proof
  • ...and 8 more