# Probabilistic finiteness properties for profinite groups

## TL;DR

This paper explores probabilistic finiteness conditions in profinite groups, introducing new concepts like PFG and PFP_n, and relates them to existing finiteness properties through module theory and projective covers.

## Contribution

It defines and characterizes probabilistic finiteness properties for profinite groups, extending classical finiteness concepts with a probabilistic perspective.

## Key findings

- Introduces PFG and PFP_n properties for profinite groups.
- Provides characterizations of these properties using module theory.
- Establishes relationships between probabilistic and classical finiteness conditions.

## Abstract

We introduce various probablistic finiteness conditions for profinite groups related to positive finite generation (PFG). We investigate completed group rings which are PFG as modules, and use this to answer a question of Kionke and the second author on positively finitely related groups. Using the theory of projective covers, we define and characterise a probabilistic version of the $\mathrm{FP}_n$ property for profinite groups, called $\mathrm{PFP}_n$. Finally, we prove how these conditions are related to previously defined finiteness conditions and each other.