# Finiteness Properties and Profinite Completions

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 $F \ell$.

Author: Alexander Lubotzky

Source: https://archive.org/