On virtual properties and group extensions
Let \(C\) be a class of groups which is closed with respect to isomorphisms and subgroups of finite Index, and containing the trivial group 1. A group which contains a \(C\)-group of finite Index is called a virtual \(C\)-group. We investigate whether the class \(\mathrm{v}C\) of all virtual \(C\)-groups is extension closed. We construct examples showing that the following classes are not closed with respect to group extensions nor amalgamated free products: the virtual duality groups, the classes \(\mathrm{v}FP\), \(FH\), the groups of virtually finite cohomology dimension, the virtually torsion-free groups. A closure criterion is derived in order to establish results like: Let \(C\) denote one of the following classes: The surface groups, the finitely generated groups \(G\) with non-trivial centre and c\(\mathrm{cd}\;G\leq2\), all subgroups of the rationals. Then the virtual poly-\(C\)-groups are extension closed. Let \(C\) be the class of all solvable groups of finite cohomology dimension or the class of all solvable duality groups, then the class \(\mathrm{v}C\) is extension closed.
scientific article; zbMATH DE number 3558086