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Title: There are only finitely many infranilmanifolds under each nilmanifold - a new proof
Authors: Dekimpe, Karel * ×
Igodt, Paul G. *
Malfait, Wim * #
Issue Date: Sep-1994
Publisher: Elsevier science bv
Series Title: Indagationes mathematicae-new series vol:5 issue:3 pages:259-266
Abstract: Infra-nilmanifolds are a natural generalization of the flat Riemannian manifolds. Together they share the property of being completely determined (up to a well understood diffeomorphism) by their fundamental group. In the case of the flat Riemannian manifolds this goes back to the well known three theorems of Bieberbach. The first two theorems of Bieberbach were generalized for the case of infra-nilmanifolds in [1] and [7] respectively. In [6] Kyung Bai Lee announced a generalization of the third Bieberbach theorem. We will show here, by means of an example, that the proof of this theorem, as presented there, contains a counting principle which is incorrect. However, this will not reject the theorem: in fact, we propose a completely different proof of the statement using a deep result on conjugacy classes of subgroups of arithmetic groups. Moreover, a key observation in [6], most likely allows a more useful approach to an explicit classification of these manifolds as already has been shown in [4].
URI: 
ISSN: 0019-3577
Publication status: published
KU Leuven publication type: IT
Appears in Collections:Mathematics, Campus Kulak Kortrijk
* (joint) first author
× corresponding author
# (joint) last author

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