The \(\Sigma^2\)-conjecture for metabelian groups: the general case.
Let \(m\) be a positive integer. Let \(G\) be a group and \(K(G,1)\) a complex with finite \(m\)-skeleton. A character \(\chi\colon G\to\mathbb{R}\) gives rise to a height function \(h\colon\overline X\to\mathbb{R}\) on the universal covering of \(X\). The geometric invariant \(\Sigma^m(G)\), consisting of the set of classes of characters under some special equivalence using these height functions, was introduced by Bieri and Strebel (1980) on finitely generated metabelian groups. There is a connection with homotopy groups of high dimensions. It was shown by them that \(\Sigma^1\) contains the information as to whether \(G\) is finitely presented. The \(\Sigma^m\)-conjecture says that for finitely generated metabelian \(FP_m\)-groups \(\Sigma^m(G)\) can be obtained from \(\Sigma^1(G)\) by a simple process. -- The authors prove the \(\Sigma^2\)-conjecture.
scientific article; zbMATH DE number 2055677
scientific article