Finitely presented groups with transcendental spectral radius

Abstract

We provide examples of groups with transcendental spectral radius. (1) 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. (2) We also construct a finitely generated example with decidable word problem, using classical small-cancellation theory. Along the way, we prove that $C^{\prime }\left(\frac{1}{6}\right)$ groups satisfy the rapid decay property and deduce some properties on their spectral radii of independent interest.