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Algebraic hulls of solvable groups and exponential iterated integrals on solvmanifolds. (English) Zbl 1280.22013
Let \(G\) be a simply connected Lie group with Lie algebra \(\mathfrak{g}\), and \(\Gamma\) a cocompact discrete subgroup of \(G\). If \(G\) is nilpotent, then Chen’s (closed) iterated integrals induced from \(\bigwedge \mathfrak{g}_\mathbb{C}^\ast\) represent the coordinate ring of the Malcev completion of \(\Gamma\). In this paper, a generalized case is discussed where \(G\) is a solvable Lie group and (hence) \(\Gamma\) is a torsion-free polycyclic group. The author applies the theory of C. Miller [Topology 44, No. 2, 351–373 (2005; Zbl 1149.57315)] and shows that the coordinate ring of the algebraic hull of \(\Gamma\) is represented by Miller’s (closed) exponential iterated integrals induced from a \(\mathbb{Z}\)-lattice of \(\mathfrak{g}_\mathbb{C}^\ast\) and \(\bigwedge \mathfrak{g}_\mathbb{C}^\ast\).
MSC:
22E25 Nilpotent and solvable Lie groups
14H30 Coverings of curves, fundamental group
55P62 Rational homotopy theory
20F99 Special aspects of infinite or finite groups
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