Finite axiomatizability of the rank and the dimension of a pro-$\pi$ group

Martina Conte and Benjamin Klopsch

Vol. 328 (2024), No. 2, 255–273

DOI: 10.2140/pjm.2024.328.255

Abstract

The Prüfer rank $\mathrm{rk}<!--FUNCTION\; APPLICATION-->(G)$ of a profinite group $G$ is the supremum, across all open subgroups $H$ of $G$, of the minimal number of generators $d<!--FUNCTION\; APPLICATION-->(H)$. It is known that, for any given prime $p$, a profinite group $G$ admits the structure of a $p$-adic analytic group if and only if $G$ is virtually a pro-$p$ group of finite rank. The dimension $\dim <!--FUNCTION\; APPLICATION-->G$ of a $p$-adic analytic profinite group $G$ is the analytic dimension of $G$ as a $p$-adic manifold; it is known that $\dim <!--FUNCTION\; APPLICATION-->G$ coincides with the rank $\mathrm{rk}<!--FUNCTION\; APPLICATION-->(U)$ of any uniformly powerful open pro-$p$ subgroup $U$ of $G$.

Let $\pi$ be a finite set of primes, let $r\in \mathbb{N}$ and let $r={(r_p)}_{p\in \pi },d={(d_p)}_{p\in \pi }$ be tuples in $\{0,1,\dots <!--FUNCTION\; APPLICATION-->,r\}$. We show that there is a single sentence ${\sigma }_{\pi ,r,r,d}$ in the first-order language of groups such that for every pro-$\pi$ group $G$ the following are equivalent: (i) ${\sigma }_{\pi ,r,r,d}$ holds true in the group $G$, that is, $G\vDash {\sigma }_{\pi ,r,r,d}$; (ii) $G$ has rank $r$ and, for each $p\in \pi$, the Sylow pro-$p$ subgroups of $G$ have rank $r_p$ and dimension $d_p$.

Loosely speaking, this shows that, for a pro-$\pi$ group $G$ of bounded rank, the precise rank of $G$ as well as the ranks and dimensions of the Sylow subgroups of $G$ can be recognized by a single sentence in the basic first-order language of groups.

In memory of Avinoam Mann

Keywords

pro-$\pi$ group, $p$-adic analytic group, finite nilpotent group, Prüfer rank, dimension, finite axiomatizability

Mathematical Subject Classification

Primary: 20E18

Secondary: 03C98, 20A15, 20D15, 20D20, 22E20

Milestones

Received: 14 April 2023

Revised: 6 February 2024

Accepted: 30 March 2024

Published: 30 April 2024