An uncountable family of 3-generated groups with isomorphic profinite completions

TL;DR

This paper constructs an uncountable family of 3-generated residually finite groups with identical profinite completions, demonstrating that word growth rate is not determined by profinite properties.

Contribution

It introduces a novel uncountable family of groups with isomorphic profinite completions and shows that word growth rate is not a profinite invariant.

Findings

Constructed uncountable family of groups with identical profinite completions

Proved word growth rate is not a profinite property

Provided new insights into the relationship between group properties and their profinite completions

Abstract

We construct an uncountable family of 3-generated residually finite just-infinite groups with isomorphic profinite completions. We also show that word growth rate is not a profinite property.