Groups In O-minimal Structures Research Articles

Strongly minimal groups in o-minimal structures

We prove Zilber's Trichotomy Conjecture for strongly minimal expansions of two-dimensional groups, definable in o-minimal structures: Theorem. Let M be an o-minimal expansion of a real closed field, (G;+) a 2-dimensional group definable in M, and D = (G;+,...) a strongly minimal structure, all of whose atomic relations are definable in M. If D is not locally modular, then an algebraically closed field K is interpretable in D, and the group G, with all its induced D-structure, is definably isomorphic in D to an algebraic K-group with all its induced K-structure.

External definability and groups in NIP theories

We prove that many properties and invariants of definable groups in NIP theories (i.e. theories without the independence property), such as definable amenability, G / G 00 , etc., are preserved when passing to the theory of the Shelah expansion by externally definable sets, M ext , of a model M. In the light of these results, we continue the study of the ‘definable topological dynamics’ of groups in NIP theories. In particular, we prove the Ellis group conjecture relating the Ellis group to G / G 00 in some new cases, including definably amenable groups in o-minimal structures.

Topology of definable abelian groups in o-minimal structures

In this note, we prove that every definably connected, definably compact abelian definable group G in an o-minimal expansion of a real closed field with dim(G)≠4 is definably homeomorphic to a torus of the same dimension. Moreover, in the semialgebraic case the result holds for all dimensions.

Splitting definably compact groups in o-minimal structures

AbstractAn argument of A. Borel [Bor-61, Proposition 3.1] shows that every compact connected Lie group is homeomorphic to the Cartesian product of its derived subgroup and a torus. We prove a parallel result for definably compact definably connected groups definable in an o-minimal expansion of a real closed field. As opposed to the Lie case, however, we provide an example showing that the derived subgroup may not have a definable semidirect complement.

On central extensions and definably compact groups in o-minimal structures

We prove several structural results on definable, definably compact groups G in o-minimal expansions of real closed fields such as (i) G is definably an almost direct product of a semisimple group and a commutative group, (ii) (G,⋅) is elementarily equivalent to (G/G00,⋅). We also prove results on the internality of finite covers of G in an o-minimal environment, as well as deducing the full compact domination conjecture for definably compact groups from the semisimple and commutative cases which were already settled.These results depend on key theorems about the interpretability of central and finite extensions of definable groups, in the o-minimal context. These methods and others also yield interpretability results for universal covers of arbitrary definable real Lie groups.

Cohomology of groups in o-minimal structures: acyclicity of the infinitesimal subgroup

AbstractBy recent work on some conjectures of Pillay, each definably compact group in a saturated o-minimal structure is an extension of a compact Lie group by a torsion free normal divisible subgroup, called its infinitesimal subgroup. We show that the infinitesimal subgroup is cohomologically acyclic. This implies that the functorial correspondence between definably compact groups and Lie groups preserves the cohomology.

Type-definable and invariant groups in o-minimal structures

AbstractLet M be a big o-minimal structure and G a type-definable group in Mn. We show that G is a type-definable subset of a definable manifold in Mn that induces on G a group topology. If M is an o-minimal expansion of a real closed field, then G with this group topology is even definably isomorphic to a type-definable group in some Mk with the topology induced by Mk. Part of this result holds for the wider class of so-called invariant groups: each invariant group G in Mn has a unique topology making it a topological group and inducing the same topology on a large invariant subset of the group as Mn.

On torsion-free groups in o-minimal structures

On torsion-free groups in o-minimal structures

Locally definable groups in o-minimal structures

In this paper we develop the theory of locally definable groups in o-minimal structures generalizing in this way the theory of definable groups.

Covers of groups definable in o-minimal structures

We develop in this paper the theory of covers for locally definable groups in o-minimal structures. ∗With partial support from the EPSRC grant GR/M66332 and the FCT grant SFRH/BPD/6015/2001 while a post-doctoral research fellow at Oxford University and CMAF-Universidade de Lisboa. MSC: 03C64; 20E99.

Simple algebraic and semialgebraic groups over real closed fields

We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. InDefinably simple groups in o-minimal structures, we showed that every such group is a semialgebraic group over a real closed field. Our main result here, stated in a model theoretic language, is that every such group is either bi-interpretable with an algebraically closed field of characteristic zero (when the group is stable) or with a real closed field (when the group is unstable). It follows that every abstract isomorphism between two unstable groups as above is a composition of a semialgebraic map with a field isomorphism. We discuss connections to theorems of Freudenthal, Borel-Tits and Weisfeiler on automorphisms of real Lie groups and simple algebraic groups over real closed fields.

Definably simple groups in o-minimal structures

Let G = ⟨ G , ⋅ ⟩ \mathbb {G}=\langle G, \cdot \rangle be a group definable in an o-minimal structure M \mathcal {M} . A subset H H of G G is G \mathbb {G} -definable if H H is definable in the structure ⟨ G , ⋅ ⟩ \langle G,\cdot \rangle (while definable means definable in the structure M \mathcal {M} ). Assume G \mathbb {G} has no G \mathbb {G} -definable proper subgroup of finite index. In this paper we prove that if G \mathbb {G} has no nontrivial abelian normal subgroup, then G \mathbb {G} is the direct product of G \mathbb {G} -definable subgroups H 1 , … , H k H_1,\ldots ,H_k such that each H i H_i is definably isomorphic to a semialgebraic linear group over a definable real closed field. As a corollary we obtain an o-minimal analogue of Cherlin’s conjecture.

Definable homomorphisms of abelian groups in o-minimal structures

We investigate the group H of definable homomorphisms between two definable abelian groups A and B, in an o-minimal structure N . We prove the existence of a “large”, definable subgroup of H . If H contains an infinite definable set of homomorphisms then some definable subgroup of B (equivalently, a definable quotient of A) admits a definable multiplication, making it into a field. As we show, all of this can be carried out not only in the underlying structure N but also in any structure definable in N .

Sharply n-transitive groups in o-minimal structures

Sharply n-transitive groups in o-minimal structures