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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.
##### 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