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13.36 (1995)

Partially Solved

For a finitely generated pro-$p$-group $G$ set $a_n(G) = \dim_{\mathbb{F}_p} I^n/I^{n+1}$, where $I$ is the augmentation ideal of the group ring $\mathbb{F}_p[[G]]$. We define the growth of $G$ to be the growth of the sequence $\{a_n(G)\}_{n\in\mathbb{N}}$.
$\qquad$ a) If the growth of $G$ is exponential, does it follow that $G$ contains a free pro-$p$-subgroup of rank 2?
$\qquad$ b) (A. Lubotzky, A. Shalev). Is the growth of $G$ exponential if $G$ contains a finitely generated closed subgroup of exponential growth?
$\qquad$ c) Do there exist pro-$p$-groups of finite cohomological dimension which are not $p$-adic analytic, and whose growth is slower than an exponential one?

Contributor: O. V. Mel’nikov

Progress

b) Not necessarily. Let $G$ be the Nottingham group over $\mathbb{F}_p$. The growth of $c_n(G) := \log_p |G : \omega_n(G)|$, where $\{\omega_n(G)\}$ is the Zassenhaus filtration, is linear, so by Quillen’s theorem the growth of $G$ is subexponential. But $G$ has a 2-generator free pro-$p$ subgroup (R. Camina, J. Algebra, 196 (1997), 101–113) with exponential growth. (M. Ershov, Letter of 19.10.2009.)

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