Explicit universal minimal constants for polynomial growth of groups

Shalom and Tao showed that a polynomial upper bound on the size of a single, large enough ball in a Cayley graph implies that the underlying group has a nilpotent subgroup with index and degree of polynomial growth both bounded effectively. The third and fourth authors proved the optimal bound on the degree of polynomial growth of this subgroup, at the expense of making some other parts of the result ineffective. In the present paper we prove the optimal bound on the degree of polynomial growth without making any losses elsewhere. As a consequence, we show that there exist explicit positive numbers $\varepsilon_d$ such that in any group with growth at least a polynomial of degree $d$, the growth is at least $\varepsilon_dn^d$. We indicate some applications in probability; in particular, we show that the gap at $1$ for the critical probability for Bernoulli site percolation on a Cayley graph, recently proven to exist by Panagiotis and Severo, is at least $\exp\bigl\{-\exp\bigl\{17 \exp\{100 \cdot 8^{100}\}\bigr\}\bigr\}$.