# Finiteness properties of groups

Lately, I have been concerned with a number of finiteness properties of groups. Everyone is probably familiar with finite generation and finite presentation, but there are many more. We will define several types, and discuss the relationships between them. The first types of finiteness one may care about is the finiteness of a corresponding Eilenberg-MacLane space.

Definition: A group $$G$$ is of type $$F_n$$ if it has a $$K(G,1)$$ with finite $$n$$ skeleton.

Definition: A group $$G$$ is of type $$F$$ if