×

zbMATH — the first resource for mathematics

Finite presentability and Heisenberg representations. (English) Zbl 0742.20036
Groups, Vol. 1, Proc. Int. Conf., St. Andrews/UK 1989, Lond. Math. Soc. Lect. Note Ser. 159, 52-64 (1991).
[For the entire collection see Zbl 0722.00007.]
This is an attempt to generalize the geometric invariant \(\Sigma\) of a finitely generated group \(G\) of R. Bieri, W. D. Neumann and R. Strebel [Invent. Math. 90, 451-477 (1987; Zbl 0642.57002)]. Recall that \(\Sigma\) is an open subset of the sphere \(S(G)=(\hbox{Hom}(G/G',\mathbb{R})- \{0\})/\mathbb{R}_{>0}\), where we are taking the orbit space of positive real action on the real vector space \(\hbox{Hom}(G/G',\mathbb{R})\).
The author suggests a generalization of this: instead of considering homomorphisms \(G/G'\to \mathbb{R}\) one should fix a natural torsion-free nilpotent quotient \(G\twoheadrightarrow H_ 0\), embed it as a discrete subgroup into a real Lie-group \(H\) and consider the space \(\Omega(G)=(L(H)^*-\{0\})/\mathbb{R}_{>0}\times H\), where \(L(H)^*\) is the dual of the Lie-algebra of \(H\). He makes his suggestion explicit by taking \(H_ 0\subseteq H\) to be the embedding of the integral into the real Heisenberg group. He introduces a generalized geometric invariant \(\Sigma'\subseteq \Omega(G)\) and proves the
Theorem. If \(G\) is finitely presented and contains no free subgroups of rank 2 then \(\Omega=\Sigma'\cup(-\Sigma')\), where \(-\Sigma'\) is the image of \(\Sigma'\) under the antipodal map.
This generalizes Theorem 5.1 of the B-N-S-paper above and so — potentially — provides a stronger necessary condition for a group \(G\) to be finitely presented. To prove that the new condition is actually stronger would require to construct a specific example.
MSC:
20F65 Geometric group theory
20F05 Generators, relations, and presentations of groups
20F16 Solvable groups, supersolvable groups
22E40 Discrete subgroups of Lie groups
57M05 Fundamental group, presentations, free differential calculus