Generalized cohomological dimension
The cohomological dimension cd of a group G is an invariant which is extremely sensitive to the presence of torsion. There are several ways to reduce this sensitivity. An easy alternative is to use \(cd_{{\mathbb{Q}}}G\) instead of \(cd_{{\mathbb{Z}}}G\) or the virtual cohomological dimension vcd G. In the paper under review, a new invariant is defined and studied. The generalized cohomological dimension is introduced as \[ \underline{cd} G=\sup \{k| \quad Ext^ k_{{\mathbb{Z}}G}(M,F)\neq 0,\quad M\quad {\mathbb{Z}}-free,\quad F\quad {\mathbb{Z}}G-free\}. \] The new invariant has properties reminiscent of vcd, e.g. If \underbar{cd} G\(=0\) then G is torsion and if G is finite, then \underbar{cd} G\(=0\). (Conjecture: \underbar{cd} G\(=0\) iff G is finite.) Theorem: If G is of type \(FP_ 2\) and \underbar{cd} G\(=0\) then G is finite. - A result on the behaviour of \underbar{cd} with respect to extensions is used in Theorem: Let G be solvable with \(| G| =\aleph_ k\), h(G) the Hirsch number of G and L the length of a solvable series of G, then \underbar{cd} \(G\leq h(G)+L(k+1)\). - Dealing with discrete subgroups of Lie groups, the author is able to show Theorem: Let G be a Lie group with finitely many connected components and \(\Gamma <G\) discrete. Let \(K<G\) be maximal compact and \(X=G/K={\mathbb{R}}^ d\), then \underbar{cd} \(\Gamma\leq d\). - It would be interesting to know the precise difference between \(cd_{{\mathbb{Q}}}G\) and \underbar{cd} G.
scientific article; zbMATH DE number 3939590
scientific article