## ON CENTRAL EXTENSIONS AND DEFINABLY COMPACT GROUPS IN O-MINIMAL STRUCTURES

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@MISC{Hrushovski_oncentral,

author = {Ehud Hrushovski and Anand Pillay},

title = {ON CENTRAL EXTENSIONS AND DEFINABLY COMPACT GROUPS IN O-MINIMAL STRUCTURES},

year = {}

}

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### Abstract

Abstract. We prove several structural results on 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/G 00, ·). 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. 1. Introduction and

### Citations

247 | den Dries. Tame topology and o-minimal structures, volume 248 - van - 1998 |

135 | Lie Groups and Algebraic Groups - Onishchik, Vinberg - 1990 |

65 | Model theory and modules - Prest |

41 |
On groups and fields definable in o-minimal structures
- Pillay
- 1988
(Show Context)
Citation Context ...cture N . In general, M is an o-minimal structure, with M is its universe and as a rule our groups G, ˜ G, H etc. are all definable in M (again with parameters). However, in order to use results from =-=[23]-=- about topology of groups we add the extra assumption that in the structure M, every such group is definably isomorphic to a group whose universe is a subset of M n . All main results assume that M ha... |

34 | Definable compactness and definable subgroups of ominimal groups
- Peterzil, Steinhorn
- 1999
(Show Context)
Citation Context ...rameters). If G is stable then it is a linear algebraic group over a definable algebraically closed field K. Because K is a definable algebraically closed field in the o-minimal structure M, then, by =-=[22]-=-, a maximal real closed subfield R ⊆ K is definable in M and we have K = R( √ −1). Since G is a linear algebraic group over K, we may assume that G ⊆ K ℓ for some ℓ and that its group-topology agrees ... |

28 | Definably compact abelian groups
- Edmundo, Otero
- 2004
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Citation Context ...S1) into S2 ∩ C/S1 ∩ C and hence |T or(S2/S1)| � [S2 ∩ C : S1 ∩ C]. If [S2 : S1] is infinite then dimS1 < dimS2 and therefore S2/S1 is a definably compact group of positive dimension. It follows from =-=[9]-=- that T or(S2/S1) is infinite and therefore, by the inequality above, so is [S2 ∩ C : S1 ∩ C]. If S2/S1 is finite then all its elements are torsion and therefore, by the same inequality we have S2/S1 ... |

20 |
Definably simple groups in o-minimal structures
- Peterzil, Pillay, et al.
- 2000
(Show Context)
Citation Context ...tity of a definable group G. We denote by G its Lie algebra whose underlying R-vector space is the tangent space of G at e, Te(G) . We recall some facts about groups and Lie algebras, as presented in =-=[18]-=-. Assume that G is definably connected. To every definable subgroup H ⊆ G there is an associated Lie subalgebra h ⊆ G. The subgroup H is normal in G if and only if h is an ideal in G (see [18, Theorem... |

19 |
O-minimal fundamental group, homology and manifolds
- Berarducci, Otero
- 2002
(Show Context)
Citation Context ...ion of work by Edmundo ([6]) and Edmundo-Eleftheriou ([7]) on universal covers and local definability in an o-minimal setting, as well as work on definable fundamental groups by Berarducci and Otero (=-=[3]-=-). We include the material because we could not find it precisely stated in the literature. In any case thanks to Edmundo for his explanations to us of results implicit in his work, some of which we r... |

16 | Groups and rings definable in o-minimal expansions of real closed fields - Otero, Peterzil, et al. - 1996 |

10 | Solvable groups definable in o-minimal structures
- Edmundo
- 2003
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Citation Context ...of A × G 2r by an M-definable equivalence relation ∼ which is itself the pre-image of equality under the continuous map j. The isomorphism h ˜ G : ˜ G ′ → G is just the map induced by j. Note that by =-=[5]-=-, the structure M has definable choice functions for subsets of A × G, hence there exists a definable set of representatives X ⊆ A ×G 2r and a definable bijection α : ˜ G ′ → X. By the definition of t... |

5 |
On NIP and invariant measures, preprint
- Hrushovski, Pillay
- 2009
(Show Context)
Citation Context ... EXTENSIONS 31 The conjecture was proved in [13] for G with “very good reduction”, and by part (ii) of Theorem 4.4 of the current paper, this is the case for semisimple definably connected groups. In =-=[14]-=- compact domination was proved for G commutative. With 6.4 we know that arbitrary G (definably compact, definably connected) almost splits into its semisimple and abelian parts, and one would expect t... |

4 |
Hurewicz theorems for definable groups, Lie groups and their cohomologies, Preprint (http://www.ciul.ul.pt/ edmundo
- Edmundo, Jones, et al.
(Show Context)
Citation Context ... sets of the form Xi \ Yi, so constructible. With this ends the proof of the claim and of Lemma 4.2 � Part (ii) of the theorem below follows from a general result by Edmundo, Jones and Peatfield, see =-=[8]-=-. Theorem 4.4. If ˜ G is semisimple and definably connected then (i) 〈 ˜ G, ·〉 is bi-interpretable with 〈 ˜ G/Z( ˜ G), ·〉, after naming a parameter ¯ b from ˜G/Z( ˜ G). The parameter can be chosen in ... |

4 |
Returning to semi-bounded sets
- Peterzil
(Show Context)
Citation Context ...ed semi-bounded if there is no definable bijection between bounded and unbounded intervals. There are three different possibilities for an o-minimal expansion M of an ordered group (see discussion in =-=[17]-=-): 1. T h(M) is linear, i.e. M is elementarily equivalent to an ordered reduct of an ordered vector space over an ordered division ring.sCENTRAL EXTENSIONS 15 2. T h(M) is not linear and not semi-boun... |

3 | Zero-groups and maximal tori, in Logic Colloquium 2004 A. Andretta, et al. (eds - Berarducci |

3 |
The Last Word on Elimination of Quantifiers in Modules
- Gute, Reuter
- 1990
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Citation Context ...jective group homomorphism φ : A → B which is the identity map on B and in addition sends every 0-definable S ⊆ A n onto S ∩ B n . (We call such a φ a homomorphic retract). Proof. (1)(i) is proved in =-=[11]-=-. 1(ii) can be extracted from the proof of the quantifier elimination result in [11], in exactly the same way as the analogous statement for theories of modules is deduced from the proof of pp elimina... |

2 |
groups definable in o-minimal structures
- Linear
(Show Context)
Citation Context ...not every definably connected group in an o-minimal structure is elementarily equivalent to a real Lie group which is definable in an o-minimal structure. This is a small variation of an examplesfrom =-=[20]-=-, so we will be brief: CENTRAL EXTENSIONS 17 Example Let M = 〈R, <, +, ·, exp〉 be a nonstandard model of theory of the real exponential field, and let α ∈ R be element greater than all natural numbers... |

1 | spectra, infinitesimal subgroups and cohomology - O-minimal |

1 |
of groups definable in o-minimal structures
- Covers
(Show Context)
Citation Context ...her than ˜ G). � We end this diversion into extensions of definable real Lie groups by considering topological covers and related central extensions. This is really an application of work by Edmundo (=-=[6]-=-) and Edmundo-Eleftheriou ([7]) on universal covers and local definability in an o-minimal setting, as well as work on definable fundamental groups by Berarducci and Otero ([3]). We include the materi... |

1 |
Edmundo and Pantelis Eleftheriou, The universal covering homomorphism in o-minimal expansions of groups
- Mario
(Show Context)
Citation Context ...iversion into extensions of definable real Lie groups by considering topological covers and related central extensions. This is really an application of work by Edmundo ([6]) and Edmundo-Eleftheriou (=-=[7]-=-) on universal covers and local definability in an o-minimal setting, as well as work on definable fundamental groups by Berarducci and Otero ([3]). We include the material because we could not find i... |

1 |
Eleftheriou and Sergei Starchenko, Groups definable in ordered vector spaces over ordered division rings
- Pantelis
(Show Context)
Citation Context ...on-free it is elementarily equivalent to Rdim G0 . The group G1 = G/G0 is a definably compact, definably connected group and therefore by [9] (and, in the case that M expands an ordered group also by =-=[10]-=- and [17]), T or(G1) is isomorphic to the torsion group of the real torus Tdim G1 . It follows (say, by 11.2) that G1 is elementarily equivalent to the semialgebraic T dim G1 and G is elementarily equ... |

1 | The structure of compact groups, de Gruyter - Karl, Sidney - 1998 |

1 |
algebraic and semialgebraic groups over real closed fields
- Simple
(Show Context)
Citation Context ...t H can be defined over Ralg). (2) If G is definably simple then it is either bi-interpretable, over parameters, with a real closed field or with an algebraically closed field of characteristic zero, =-=[19]-=-. (3) If G is definably connected and semisimple then Z(G) is finite and G/Z(G) is definably isomorphic in M to the direct product of definably simple groups (see [18, 4.1]). (4) If G is definably com... |