The geometric invariants of direct products of virtually free groups
Let \(G\) be a group and let \(V(G)\) be the real vector space \(\text{Hom}(G;\mathbb{R})\). For every natural integer \(m\), there is a homological (respectively homotopical) invariant of \(G\), denoted by \(\Sigma^m(G,\mathbb{Z})\) (resp. \(\Sigma^m(G))\), which is defined as a certain conical subset of \(V(G)\). These invariants have been studied and computed for certain groups by R. Bieri and B. Renz. (We note that for the definition of the invariant \(\Sigma^m(G)\), the group is required to be of type \(F_m\), that is, that an Eilenberg-MacLane complex \(K(G,1)\) of \(G\) with finite \(m\)-skeleton exists). It is an open problem as to whether the two invariants coincide in the most general situation. In this paper, the author proves that in the case where \(G\) is equal to a finite direct product of finitely generated virtually free groups, we have, for every \(m\), \(\Sigma^m(G,\mathbb{Z})=\Sigma^m(G)\), and he gives a formula for this invariant, that is to say, a formula which describes the corresponding subset of \(V(G)\).
finite direct product of finitely generated virtually free groups
scientific article; zbMATH DE number 590897
scientific article