A $p$-group with positive Rank Gradient

Details A $p$-group with positive Rank Gradient A $p$-group with positive Rank Gradient

2011-05-09

arxiv.org/abs/1105.1631v1

Mathematics

Group Theory

Scientific paper

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.

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