6.3 (1978)

A group $G$ is called of type $(FP)_\infty$ if the trivial $G$-module $\mathbb{Z}$ has a resolution by finitely generated projective $G$-modules. The class of all groups of type $(FP)_\infty$ has a couple of excellent closure properties with respect to extensions and amalgamated products (R. Bieri, Homological dimension of discrete groups, Queen Mary College Math. Notes, London, 1976). Is every periodic group of type $(FP)_\infty$ finite? This is related to the question whether there is an infinite periodic group with a finite presentation.