Finiteness Properties and Profinite Completions

In this note we show that various (geometric/homological) finiteness properties are not profinite properties. For example for every $1 \le k, \ell \le \bbn$,thereexisttwofinitelygeneratedresiduallyfinitegroups$\Ga_1$and$\Ga_2$withisomorphicprofinitecompletions,suchthat$\Ga_1$ is strictly of type $F_k$ and ${\text{Ga}}_2$ of type $F_{\mathrm{\ell }}$.