A semi-linear group which is not affine

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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

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