Certifying Anosov representations

TL;DR

This paper introduces practical finite criteria and an algorithm to efficiently verify whether a finitely generated subgroup of SL(d,R) or SL(d,C) is projective Anosov, significantly reducing computational complexity.

Contribution

The authors develop new finite criteria for certifying Anosov representations and implement an efficient algorithm to verify these criteria, improving upon previous exhaustive methods.

Findings

Successfully verified Anosov property for a genus 2 surface group in SL(3,R) using words of length 8

Reduced the verification process from checking all words of length 2 million to a manageable finite set

Demonstrated practical applicability of the criteria with concrete examples

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.