Finitely presented groups with transcendental spectral radius

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 $C^{\prime }(1/6)$ groups satisfy the Rapid Decay property, and deduce some properties on their spectral radii of independent interest.

v3: Incorporated comments by referees. Added Section 6.3. (Numbering changed!) v2: Expended Theorem A, added Theorem B, an appendix and a few questions