On groups of finite upper rank

TL;DR

This paper investigates whether finitely generated groups with finite upper p-ranks for all primes necessarily have bounded upper p-ranks, highlighting an open problem in the case of soluble groups.

Contribution

It explores the relationship between finite upper p-ranks and their boundedness in finitely generated groups, focusing on unresolved cases for soluble groups.

Findings

Established the equivalence between finite upper rank and boundedness across primes.

Identified the open problem for soluble groups regarding boundedness of upper p-ranks.

Discussed deep connections between group quotients and rank properties.

Abstract

The `upper rank' of a group is the supremum of the (Pr\"{u}fer) ranks of its finite quotients, and for a prime $p$, the `upper $p$-rank' is the supremum of the sectional $p$-ranks of those quotients. The former is finite if and only if the latter are finitely bounded as $p$ ranges over all primes (a deep fact). Here we discuss the question: if the upper $p$-ranks of a finitely generated group $G$ are all finite, are they necessarily bounded? The case where $G$ is a soluble group is still an open problem.