Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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
$\exp\bigl\{-\exp\bigl\{17 \exp\{100 \cdot 8^{100}\}\bigr\}\bigr\}$.

21
The categories of this paper are: adap-org
31
The categories of this paper are: ao-sci
15
The categories of this paper are: nucl-ex
7
The categories of this paper are: hep-th
4
The categories of this paper are: cond-mat
22
The categories of this paper are: chao-dyn
13
The categories of this paper are: quant-ph
5
The categories of this paper are: gr-qc
25
The categories of this paper are: patt-sol
14
The categories of this paper are: math-ph
11
The categories of this paper are: hep-lat
34
The categories of this paper are: chem-ph
8
The categories of this paper are: hep-ex
32
The categories of this paper are: atom-ph
18
The categories of this paper are: econ
17
The categories of this paper are: q-fin