Constructing uncountably many groups with the same profinite completion

TL;DR

This paper constructs two uncountable families of finitely generated residually finite groups, one soluble and one branch, all sharing the same profinite completion, highlighting the diversity of groups with identical profinite properties.

Contribution

It introduces the first known uncountable families of finitely generated residually finite groups with identical profinite completions, including soluble and branch groups.

Findings

Uncountably many groups share the same profinite completion.

Construction of families includes soluble and branch groups.

Demonstrates diversity in groups with identical profinite properties.

Abstract

We present two uncountable families of finitely generated residually finite groups all having the same profinite completion. One consists of soluble groups, the other of branch groups.