Lattices in Locally Definable Subgroups of <Rn,+>

Let M be an o-minimal expansion of a real closed field R. We define the notion of a lattice in a locally definable group and then prove that every connected, definably generated subgroup of 〈R,+〉 contains a definable generic set and therefore admits a lattice. The goal of this note is to re-formulate some problems which appeared in [4], introduce the notion of a lattice in a locally definable group (a notion which also appeared in that paper, but not under this name) and establish connections between various related concepts. Finally, we return to the main conjecture from [4]: Every locally definable connected, abelian group, which is generated by a definable set contains a definable generic set. We prove the conjecture for subgroups of 〈Rn,+〉, in the context of an o-minimal expansion M of a real closed field R. 1. Locally definable groups and lattices We first recall some definitions: Let M be an arbitrary κ-saturated ominimal structure (for κ sufficiently large). By a locally definable group we mean a group 〈U , ·〉, whose universe U = ⋃ n∈NXn, is a countable union of definable subsets of Mk, for some fixed k, and the group operation is definable when restricted to each Xm ×Xn (equivalently, to each definable subset of U × U). We say that a function f : U → Mn is locally definable if its restriction to each Xi (equivalently, to each definable subset of U) is definable. We let dimU be the maximum of dimXn, n ∈ N. While some notions treated here make sense under the more general “ ∨ -definable group” (no restriction on the number of Xi’s), we mostly work in the context of a group which is generated, as a group, by a definable subset and hence it is locally definable. Note that another related concept, that of an ind-definable group (see [6]) is identical to our definition when one further assumes that the group is a subset of a fixed Mk. As was shown in [7], every locally definable group admits a group topology. This topology agrees with the Mk-topology in neighborhoods of generic points, namely, points g ∈ U such that dim(g/A) = dim(U) (we assume here that all the Xi’s above are defined over A). We therefore obtain a definable family of neighborhoods {Ut : t ∈ T} of the identity element, such that {gUt : t ∈ T, g ∈ U} is a basis for the group topology on U . In [2] Date: February 27, 2012. 2010 Mathematics Subject Classification. 03C64, 03C68, 22B99.