Explicit universal minimal constants for polynomial growth of groups

Abstract Shalom and Tao showed that a polynomial upper bound on the size of single, large enough ball in Cayley graph implies underlying group has nilpotent subgroup with index degree growth both bounded effectively. The third fourth authors proved optimal this subgroup, at expense making some other parts result ineffective. In present paper, we prove without any losses elsewhere. As consequence, show there exist explicit positive numbers ε d \varepsilon_{d} such that, least