Finitely presented groups with transcendental spectral radius

TL;DR

This paper constructs finitely presented groups with transcendental spectral radius, linking group theory, decidability, and spectral properties, and explores related properties like exponential growth and entropy.

Contribution

It introduces new examples of finitely presented groups with transcendental spectral radius and connects decidability of the Word Problem to spectral computability.

Findings

Finitely presented groups with transcendental spectral radius are constructed.

Links between Word Problem decidability and spectral radius semi-computability are established.

Proves $C'(1/6)$ groups satisfy the Rapid Decay property.

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.