Central theorems for cohomologies of certain solvable groups. (English) Zbl 1401.20035

Kasuya, Hisashi

Trans. Am. Math. Soc. 369, No. 4, 2879-2896 (2017).

Summary: We show that the group cohomology of torsion-free virtually polycyclic groups and the continuous cohomology of simply connected solvable Lie groups can be computed by the rational cohomology of algebraic groups. Our results are generalizations of certain results on the cohomology of solvmanifolds and infra-solvmanifolds. Moreover as an application of our results, we give a new proof of the surprising cohomology vanishing theorem given by K. Dekimpe and P. Igodt [Invent. Math. 129, No. 1, 121–140 (1997; Zbl 0867.20031)].

MSC:

20F16	Solvable groups, supersolvable groups
20G10	Cohomology theory for linear algebraic groups
20J06	Cohomology of groups
22E41	Continuous cohomology of Lie groups
22E25	Nilpotent and solvable Lie groups
17B56	Cohomology of Lie (super)algebras
57T15	Homology and cohomology of homogeneous spaces of Lie groups

Keywords:

group cohomology of torsion-free virtually polycyclic group; continuous cohomology of simply connected solvable Lie group; rational cohomology of algebraic group; de Rham cohomology of solvmanifold

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