Monotone parameters on Cayley graphs of finitely generated groups

Abstract

We construct a new large family of finitely generated groups with continuum many values of the following monotone parameters: spectral radius, critical percolation, and asymptotic entropy. We also present several open problems on other monotone parameters.

In v2, Thm 1.2 is strengthened to close the gap. Also, we revised attributions, added references, expanded final and historical remarks; 19 pages