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Deformations and rigidity of lattices in solvable Lie groups. (English) Zbl 1287.22004

The main result of this paper generalizes the classical rigidity theorems of Malcev and Saito for lattices in nilpotent Lie groups and in solvable Lie groups of real type respectively.
The lattice \(\Gamma\) is called Zariski-dense in \(G\) if the algebraic closure of \(\mathrm{Ad}_G(\Gamma)\) equals the algebraic closure of \(\mathrm{Ad}_G(G)\) (here \(\mathrm{Ad}_G\) is the adjoint representation).
It is proved here that the deformation space of every Zariski-dense lattice \(\Gamma\) in a solvable Lie group \(G\) is finite and Hausdorff, if the maximal nilpotent normal subgroup of G is connected. Moreover, the cardinality of \(D(\Gamma, G)\) is uniformly bounded above by a constant depending only on the dimension of \(G\).
Also it is proved that every lattice in a solvable Lie group virtually embeds as a Zariski-dense lattice into some simply connected solvable Lie group with finite deformation space.
Some examples of solvable Lie groups \(G\) which admit Zariski-dense lattices \(\Gamma\) such that \(D(\Gamma, G)\) is countably infinite and examples where the maximal nilpotent normal subgroup of \(G\) is connected, but this \(G\) has lattices with uncountable deformation space, are given.
A lot of other interesting results connected with the notions of deformation and rigidity can be found in this paper.

MSC:

22E40 Discrete subgroups of Lie groups
22E25 Nilpotent and solvable Lie groups
53C24 Rigidity results
20F16 Solvable groups, supersolvable groups
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