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Crystallographic actions on contractible algebraic manifolds


Authors: Karel Dekimpe and Nansen Petrosyan
Journal: Trans. Amer. Math. Soc. 367 (2015), 2765-2786
MSC (2010): Primary 20H15, 20F65; Secondary 14L17, 14L30
DOI: https://doi.org/10.1090/S0002-9947-2014-06160-6
Published electronically: November 12, 2014
MathSciNet review: 3301881
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Abstract: We study properly discontinuous and cocompact actions of a discrete subgroup $ \Gamma $ of an algebraic group $ G$ on a contractible algebraic manifold $ X$. We suppose that this action comes from an algebraic action of $ G$ on $ X$ such that a maximal reductive subgroup of $ G$ fixes a point. When the real rank of any simple subgroup of $ G$ is at most one or the dimension of $ X$ is at most three, we show that $ \Gamma $ is virtually polycyclic. When $ \Gamma $ is virtually polycyclic, we show that the action reduces to an NIL-affine crystallographic action. Specializing to NIL-affine actions, we prove that the generalized Auslander conjecture holds up to dimension six and give a new proof of the fact that every virtually polycyclic group admits an NIL-affine crystallographic action.


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Additional Information

Karel Dekimpe
Affiliation: Department of Mathematics, KU Leuven, Campus Kortrijk, Kortrijk, Belgium
Email: Karel.Dekimpe@kuleuven-kortrijk.be

Nansen Petrosyan
Affiliation: Department of Mathematics, KU Leuven, Campus Kortrijk, Kortrijk, Belgium
Address at time of publication: Mathematical Sciences, University of Southampton, Highfield, Southampton SO17 1BJ, United Kingdom
Email: Nansen.Petrosyan@kuleuven-kortrijk.be, N.Petrosyan@soton.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-2014-06160-6
Keywords: Crystallographic action, algebraic manifold
Received by editor(s): September 20, 2012
Received by editor(s) in revised form: April 23, 2013
Published electronically: November 12, 2014
Additional Notes: The first author was partially supported by the Research Fund KU Leuven.
The second author was supported by the Research Fund KU Leuven and the FWO-Flanders Research Fellowship.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.