Finitely presented groups with transcendental spectral radius

2024/04/27 by Corentin Bodart, Bodart, Corentin, Osin, Denis
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO) #Mathematical Analysis and Transform Methods #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2404.17840

openalex publication_date 2024/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We provide examples of groups with transcendental spectral radius: We first construct finitely presented examples, using links between decidability of the Word Problem and semi-computability of the spectral radius. This argument extends to the exponential growth rate and the asymptotic entropy. We also construct a finitely generated example with decidable Word Problem, using classical small-cancellation theory. Along the way, we prove that C'(1/6) groups satisfy the Rapid Decay property, and deduce some properties on their spectral radii of independent interest.

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