Finitely presented groups with transcendental spectral radius

Corentin Bodart; Denis Osin

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arXiv · 2404.17840

Finitely presented groups with transcendental spectral radius

Corentin Bodart·Denis Osin

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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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Corentin Bodart, Denis Osin. 2024-04-27. Finitely presented groups with transcendental spectral radius. https://arxiv.org/abs/2404.17840

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