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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
##### Keywords:
exponential iterated integral; algebraic hull; solvmanifold
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##### References:
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