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2008

A semi-linear group which is not affine

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A semi-linear group which is not affine
In this short note we provide an example of a semi-linear group G which does not admit a semi-linear affine embedding; in other words, there is no semi-linear isomorphism between topological groups f : G G Mm, such that the group topology on G coincides with the subspace topology induced by Mm. Let M be an o-minimal structure. By "definable" we mean "definable in M" possibly with parameters. A group G = G, , eG is said to be definable if both its domain and its group operation are definable. By [Pi], we know that every definable group G Mn can be equipped with a unique definable manifold topology that makes it into a topological group. We refer to this topology as the group topology of G. It is shown in [Pi] that the group topology of G coincides with the subspace topology induced by Mn on a large subset V of G dim(G \ V ) < dim(G) . We call G affine if the group topology of G coincides with the subspace topology on (the whole of) G. Question. Is every definab...
Pantelis E. Eleftheriou
Added 08 Dec 2010
Updated 08 Dec 2010
Type Journal
Year 2008
Where APAL
Authors Pantelis E. Eleftheriou
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