Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A $p$-group with positive Rank Gradient   We construct for $d\geq 2$ and $\epsilon>0$ a $d$-generated $p$-group
$\Gamma$, which in an asymptotic sense behaves almost like a $d$-generated free
pro-$p$-group. We show that a subgroup of index $p^n$ needs $(d-\epsilon)p^n$
generators, and that the subgroup growth of $\Gamma$ satisfies
$s_{p^n}(\Gamma)>s_{p^n}(F_d^p)^{1-\epsilon}$, where $F_d^p$ is the
$d$-generated free pro-$p$-group. To do so we introduce a new invariant for
finitely generated groups and study some of its basic properties.

20
The categories of this paper are: acc-phys
21
The categories of this paper are: adap-org
10
The categories of this paper are: q-bio
23
The categories of this paper are: solv-int
29
The categories of this paper are: funct-an
26
The categories of this paper are: bayes-an
15
The categories of this paper are: nucl-ex
3
The categories of this paper are: physics
32
The categories of this paper are: atom-ph
18
The categories of this paper are: econ
24
The categories of this paper are: dg-ga
34
The categories of this paper are: chem-ph
5
The categories of this paper are: gr-qc
25
The categories of this paper are: patt-sol
27
The categories of this paper are: comp-gas
0
The categories of this paper are: hep-ph