Finiteness conditions on groups and quasi-isometries
Let \(G\) be a finitely generated group equipped with a word-metric. A major theme, which has been developed mainly by Gromov, is to study the properties of \(G\) which are invariant under quasi-isometries. A group \(G\) is said to be of type \(F_ n\) (\(n\geq 1\)) if there exists an Eilenberg-MacLane complex \(K(G,1)\) with finite \(n\)-skeleton. Condition \(FP_ n\) is a homological version. Conditions \(F_ 1\) and \(F_ 2\) are equivalent, respectively, to \(G\) being finitely generated and finitely presented. The author shows that conditions \(F_ n\) and \(FP_ n\) are quasi- isometry invariants. The paper contains also an application for properties \(F_ n\) and \(FP_ n\) for group extensions.
scientific article; zbMATH DE number 681814