Direct factors of profinite completions and decidability

TL;DR

This paper investigates the relationship between finitely presented residually finite groups and their profinite completions, demonstrating the undecidability of determining direct factors within finite index subgroups.

Contribution

It introduces the concept that certain subgroups with isomorphic profinite completions are not necessarily direct factors, and proves the undecidability of identifying such factors.

Findings

Existence of subgroups with isomorphic profinite completions that are not direct factors

No algorithm can decide if a subgroup is a direct factor of a finite index subgroup

Profinite completion properties do not guarantee subgroup splitting

Abstract

We consider finitely presented,residually finite groups $G$ and finitely generated normal subgroups $A$ such that the inclusion $A\hookrightarrow G$ induces an isomorphism from the profinite completion of $A$ to a direct factor of the profinite completion of $G$. We explain why $A$ need not be a direct factor of a subgroup of finite index in $G$; indeed $G$ need not have a subgroup of finite index that splits as a non-trivial direct product. We prove that there is no algorithm that can determine whether $A$ is a direct factor of a subgroup of finite index in $G$.