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Central theorems for cohomologies of certain solvable groups


Author: Hisashi Kasuya
Journal: Trans. Amer. Math. Soc. 369 (2017), 2879-2896
MSC (2010): Primary 20F16, 20G10, 20J06, 22E41; Secondary 22E25, 17B56, 57T15
DOI: https://doi.org/10.1090/tran/6837
Published electronically: October 12, 2016
MathSciNet review: 3592531
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Abstract: 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 Dekimpe-Igodt.


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Additional Information

Hisashi Kasuya
Affiliation: Department of Mathematics, Tokyo Institute of Technology, 1-12-1, O-okayama, Meguro, Tokyo 152-8551, Japan
Address at time of publication: Department of Mathematics, Graduate School of Science, Osaka University, Osaka 560-0043, Japan
Email: kasuya@math.titech.ac.jp, kasuya@math.sci.osaka-u.ac.jp

DOI: https://doi.org/10.1090/tran/6837
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
Received by editor(s): August 18, 2014
Received by editor(s) in revised form: May 6, 2015, and September 24, 2015
Published electronically: October 12, 2016
Article copyright: © Copyright 2016 American Mathematical Society