Finiteness Properties and Profinite Completions

TL;DR

This paper demonstrates that certain geometric and homological finiteness properties of groups are not detectable solely from their profinite completions, by constructing examples of groups with identical profinite completions but different finiteness types.

Contribution

It provides explicit examples of finitely generated residually finite groups with isomorphic profinite completions but differing in their finiteness properties, showing these properties are not profinite invariants.

Findings

Existence of groups with same profinite completion but different finiteness types

Finiteness properties are not profinite properties

Counterexamples for various finiteness conditions

Abstract

In this note we show that various (geometric/homological) finiteness properties are not profinite properties. For example for every $1 \le k, \ell \le \bbn$, there exist two finitely generated residually finite groups $\Ga_1$ and $\Ga_2$ with isomorphic profinite completions, such that $\Ga_1$ is strictly of type $F_k$ and $\Ga_2$ of type $F_\ell$.