• DOI:10.1515/jgth-2020-0202
• Corpus ID: 235732391

Explicit universal minimal constants for polynomial growth of groups

@article{Lyons2022ExplicitUM,
  title={Explicit universal minimal constants for polynomial growth of groups},
  author={Russell Lyons and Avinoam Mann and Romain Tessera and Matthew C. H. Tointon},
  journal={Journal of Group Theory},
  year={2022},
  volume={0}
}

• R. Lyons, A. Mann, +1 author Matthew C. H. Tointon
• Published 11 October 2020
• Mathematics
• Journal of Group Theory

Abstract 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 ε d \varepsilon_{d} such that, in any group with growth at least a polynomial of degree 𝑑, the growth is at least ε d ⁢ n d \varepsilon_{d}n^{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 ⁡ { - exp ⁡ { 17 ⁢ exp ⁡ { 100 ⋅ 8 100 } } } \exp\{-\exp\{17\exp\{100\cdot 8^{100}\}\}\} .