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 ⊆ M 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 M 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 definable group affine (up to definable isomorphism)? Remark 0.1. (i) An isomorphism between two topological groups is a group isomorphism which is also a topological homeomorphism. (ii) By [ElSt, Remark 2.2], the Question can be restated as follows: Given a definable group G ⊆ M, is there a definable injective map τ : G → M, m ∈ N, such that the topology on τ(G) induced by the group topology of G via τ coincides with the subspace topology on τ(G) induced by M? If yes, then such a τ is called an affine embedding of G. The Question admits an affirmative answer in case M expands a real closed field, by [BO, Proof of Lemma 10.4] and [vdD, Chapter 10, Theorem (1.8)]. In fact, these references concern affine embeddings of “abstract-definable manifolds”, and the work in [BO] also yields affine embeddings which are moreover diffeomorphisms. The original proof of embedding semi-algebraic manifolds was given in [Ro]. We present here an example of a semi-linear group which is not affine. A semi-linear group is a group definable in an ordered vector space M = 〈M, +, <, 0, {d}d∈D〉 over an ordered division ring D. Semi-linear groups were studied in [ElSt] and [El]. The main property of such an M that we use below is that every definable function f : A ⊆ M → M is piecewise-linear (PL); that is, there is a partition of A into finitely many definable sets Ai, i = 1, . . . , k, such that for Date: June 21, 2008. 2000 Mathematics Subject Classification. 03C64, 57Q35.