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Singularity of the varieties of representations of lattices in solvable Lie groups. (English) Zbl 1347.22008
Let \(\Gamma\) be a lattice in a simply connected solvable Lie group \(G\). Then the manifold \(G/\Gamma\) is an aspherical manifold with fundamental group \(\Gamma\). The purpose of this work is the study of the analytic germ \((R(\Gamma,A),1)\) at the trivial representation \(1\) for \(A\) a linear algebraic group. This can be studied by the differential graded algebra (DGA) \(A^\ast (G/\Gamma)\) of differential forms of \(G/\Gamma\). Let \(\mathfrak g\) denote a solvable Lie algebra. Then the nilshadow \(\mathfrak u\) is uniquely determined by \(\mathfrak g\). Then one has a sub-DGA \(A^\ast_\Gamma \subset A^\ast (G/\Gamma) \ltimes \mathbb C\), which induces an isomorphism in cohomolgy and such that \(A^\ast_\Gamma\) can be regarded as a sub-DGA of \(\Lambda \mathfrak u^\ast \otimes \mathbb C\). Assume that \(\mathfrak u\) is \(\nu\)-step naturally graded and that \(A\) is a linear algebraic group. Then a main theorem states that \((R(\Gamma,A),1)\) is cut out by polynomial equations of degree at most \(\nu+1\).

22E25 Nilpotent and solvable Lie groups
32B10 Germs of analytic sets, local parametrization
22E40 Discrete subgroups of Lie groups
17B55 Homological methods in Lie (super)algebras
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