A $p$-group with positive Rank Gradient

We construct for $d\geq 2$ and $ε>0$ a $d$-generated $p$-group $Γ$, 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 $Γ$ satisfies $s_{p^n}(Γ)>s_{p^n}(F_d^p)^{1-ε}$, 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.

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