Certifying Anosov representations

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.

💡 Research Summary

The paper introduces a concrete, finite‑time algorithm for certifying that a finitely generated subgroup of SL(d,ℝ) or SL(d,ℂ) is projective Anosov. While the existence of Anosov subgroups is well‑understood theoretically—thanks to the work of Labourie, Guichard‑Wienhard, and especially the local‑to‑global principle of Kapovich‑Leeb‑Porti (KLP)—the known proofs do not yield a practical computational test. In particular, the KLP algorithm requires checking an infinite family of word products; in practice the only way to guarantee termination is to verify a condition on all words up to some huge length (the author’s earlier implementation needed words of length two million).

The core contribution is a new set of finite criteria, encapsulated in Theorem 5.2, that replace the original KLP condition with a much weaker “dα‑undistorted” growth property. The author shows that if, for a given generating set, every word of length up to a modest bound L satisfies a linear lower bound on the first singular‑value gap (equivalently, on the root pseudo‑metric dα), then the same bound propagates to all longer words. This propagation relies on two technical ingredients:

1. Lemma 4.1 (Angle‑to‑Distance Formula). For two transverse points τ⁺∈ℙ^{d‑1} and τ⁻∈(ℙ^{d‑1})* and any point q in the symmetric space X=SL(d,K)/SU(d,K), the distance from q to the parallel set P(τ⁻,τ⁺) determines the visual angle ∠_q(τ⁻,τ⁺) via the exact identity
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