Abstract
AbstractWe establish the first global results for groups definable in tame expansions of o-minimal structures. Let ${\mathcal{N}}$ be an expansion of an o-minimal structure ${\mathcal{M}}$ that admits a good dimension theory. The setting includes dense pairs of o-minimal structures, expansions of ${\mathcal{M}}$ by a Mann group, or by a subgroup of an elliptic curve, or a dense independent set. We prove: (1) a Weil’s group chunk theorem that guarantees a definable group with an o-minimal group chunk is o-minimal, (2) a full characterization of those definable groups that are o-minimal as those groups that have maximal dimension; namely, their dimension equals the dimension of their topological closure, (3) as an application, if ${\mathcal{N}}$ expands ${\mathcal{M}}$ by a dense independent set, then every definable group is o-minimal.
Highlights
Definable groups have been at the core of model theory for at least a period of three decades, largely because of their prominent role in important applications of the subject, such as Hrushovski’s proof of the function field Mordell–Lang conjecture in all characteristics [23]
Examples include algebraic groups and real Lie groups
In this paper we prove the first global results, whose gist is that one can recover an o-minimal group from an arbitrary definable group using only dimension-theoretic data
Summary
Definable groups have been at the core of model theory for at least a period of three decades, largely because of their prominent role in important applications of the subject, such as Hrushovski’s proof of the function field Mordell–Lang conjecture in all characteristics [23]. Let N be an expansion of an o-minimal structure M such that (a) every open definable set is definable in M, and (b) N admits a dimension function compatible with M. The influential Pila–Wilkie theorem [31] recovers a semialgebraic subset of a set X definable in an arbitrary o-minimal structure given a number-theoretic condition on X. In the example N = M, P , where P is a dense dcl-independent set, our work implies that there are no new definable groups at all This pair recently received special attention in [8] and even triggered new model-theoretic work at the general level of ‘H -structures’ [4]. Let us point out that Theorem 1.1 establishes a conjecture for definable groups stated in [15] and reformulated in [17], for the case of strongly large groups.
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