Positively finitely related profinite groups

TL;DR

This paper introduces and studies positively finitely related (PFR) profinite groups, a probabilistic class that generalizes positively finitely generated groups, with characterizations based on representation growth and structural properties.

Contribution

It defines PFR profinite groups, establishes asymptotic characterizations, and explores their structural implications, extending the theory of finitely generated profinite groups.

Findings

PFR groups are characterized by exponential representation growth.

A finitely presented profinite group is PFR iff it has polynomial maximal ideal growth.

Structural properties of PFR groups are derived from these characterizations.

Abstract

We define and study the class of positively finitely related (PFR) profinite groups. Positive finite relatedness is a probabilistic property of profinite groups which provides a first step to defining higher finiteness properties of profinite groups which generalize the positively finitely generated groups introduced by Avinoam Mann. We prove many asymptotic characterisations of PFR groups, for instance we show the following: a finitely presented profinite group is PFR if and only if it has at most exponential representation growth, uniformly over finite fields (in other words: the completed group algebra has polynomial maximal ideal growth). From these characterisations we deduce several structural results on PFR profinite groups.