Pansu–Grigorchuk gap conjectures for heat kernels, Følner functions and spectral density

Let $G$ be an amenable finitely generated group, let $\mu$ be a symmetric finitely supported probability measure generating $G$, let $M$ be the associated Markov operator, and let $P(n)$ be the return probability after $n$ steps. Let $F(n)$ be the Følner function. Write $\Delta=I-M$ and let $\mathcal N(\lambda)$ be the spectral density of $\Delta$ with respect to the von Neumann trace. Pansu–Grigorchuk's gap conjectures.

1. The function $P(n)$ either has power-rate decay or satisfies

$$P(n)\preceq e^{-\sqrt[3]{n}}.$$

2. The function $F(n)$ either has polynomial growth or grows at least exponentially.
3. The function $\mathcal N(\lambda)$ either has power decay of type $\lambda^{d/2}$ for some $d\in\mathbb N$ as $\lambda\to0$, or satisfies

$$\mathcal N(\lambda)\preceq e^{-1/\sqrt[4]{\lambda}}.$$

These are three related conjectures about asymptotic gaps for random-walk return probabilities, Følner functions and spectral densities; the paper presents them as open conjectures formulated by Pansu and the author.