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The cohomology of virtually torsion-free solvable groups of finite rank


Authors: Peter Kropholler and Karl Lorensen
Journal: Trans. Amer. Math. Soc. 367 (2015), 6441-6459
MSC (2010): Primary 20F16, 20J05, 20J06
DOI: https://doi.org/10.1090/S0002-9947-2015-06262-X
Published electronically: March 13, 2015
MathSciNet review: 3356943
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Abstract: Assume that $ G$ is a virtually torsion-free solvable group of finite rank and $ A$ is a $ \mathbb{Z}G$-module whose underlying abelian group is torsion-free and has finite rank. We stipulate a condition on $ A$ that ensures that $ H^n(G,A)$ and $ H_n(G,A)$ are finite for all $ n\geq 0$. Using this property for cohomology in dimension two, we deduce two results concerning the presence of near supplements and complements in solvable groups of finite rank. As an application of our near-supplement theorem, we obtain a new result regarding the homological dimension of solvable groups.


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Additional Information

Peter Kropholler
Affiliation: Mathematical Sciences, University of Southampton, Highfield, Southampton SO17 1BJ, United Kingdom
Email: P.H.Kropholler@southampton.ac.uk

Karl Lorensen
Affiliation: Department of Mathematics and Statistics, Pennsylvania State University, Altoona College, Altoona, Pennsylvania 16601
Email: kql3@psu.edu

DOI: https://doi.org/10.1090/S0002-9947-2015-06262-X
Received by editor(s): March 19, 2013
Received by editor(s) in revised form: April 1, 2013, and July 14, 2013
Published electronically: March 13, 2015
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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