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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.
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