A $p$-group with positive Rank Gradient

Abstract

We construct for $d\ge 2$ 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.

References in corpus (2)

• Rank gradient, cost of groups and the rank versus Heegaard genus problem
• A characterisation of large finitely presented groups

Cited by in corpus (1)

• Groups of p-deficiency one