A $p$-group with positive Rank Gradient

We construct for $d\ge 2$ and $ϵ>0$ a $d$-generated $p$-group $\mathrm{\Gamma }$, 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 $(d-ϵ)p^n$ generators, and that the subgroup growth of $\mathrm{\Gamma }$ satisfies $s_{p^n}(\mathrm{\Gamma })>s_{p^n}(F_d^p{)}^{1-ϵ}$, 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.