There are some equivalent definitions for "finiteness properties", but let's define that $G$ is a group of type $F_n$ if it is the fundamental group of a CW complex $X$ whose $n$-skeleton is finite and whose universal cover is contractible.
A claim that I see everywhere but without a proof is that a group is $F_1$ if and only if it's finitely generated, and a group is $F_2$ if and only if it's finitely presented.
Actually both directions (of both claims) are not clear to me (although the specific implication I need is the one in the title...).
In one direction I guess I should use the presentation complex but the universal cover need not be contractible. In the other direction I know that examining the $2$-skeleton of a complex is enough to determine the fundamental group of the whole complex, but does the fact that this skeleton is finite imply immediately that the fundamental group is f.p.? Or should I use contractibility of the universal cover somehow?
I'll be thankful even for a reference to a proof, I don't know why I can't find any... Is it so elementary? :)