Definably compact groups definable in real closed fields
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DEFINABLY COMPACT GROUPS DEFINABLE IN REAL CLOSED FIELDS
BY
Eliana Barriga Departmento de Matem´ aticas, Universidad de los Andes Cra 1 No 18A-10, Bogot´ a 111711, Colombia e-mail: [email protected]
ABSTRACT
We study definably compact definably connected groups definable in a sufficiently saturated real closed field R. Our main result is that for such a kind of groups G that are also abelian, there is a Zariski-connected Ralgebraic group H such that the o-minimal universal covering group of G is, up to a locally definable isomorphism, an open connected locally definable subgroup W of the o-minimal universal covering group of H(R)0 . Thus, G is definably isomorphic to the definable quotient of W by a discrete subgroup.
1. Introduction Definable groups in o-minimal structures have been intensively studied in the last three decades, and it is a field of current research. A real closed field is an ordered field elementarily equivalent to the real ordered field R. By quantifier elimination in real closed fields (Tarski–Seidenberg), the definable sets in a real closed field R are the semialgebraic sets over R. Since a real closed field is an o-minimal structure (i.e., an ordered structure for which every definable subset of its universe is a finite union of points and intervals; see, e.g., [7]), then semialgebraic groups over a real closed field can be seen as a generalization of the semialgebraic groups over the real field R, and also as a particular case of the groups definable in an o-minimal structure. Received November 4, 2018 and in revised form June 13, 2019
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E. BARRIGA
Isr. J. Math.
Let G be a group definable in an o-minimal structure. Pillay proved in [22] that G can be equipped with a unique definable manifold structure making the group into a topological group; we call this topology on G the t-topology. From now on any topological property of definable groups in an o-minimal structure refers to this t-topology. We say that a definable subset X ⊆ G is definably connected if X has no nonempty proper definable subset (t-)clopen relative to X. By [22, Corollary 2.10], there is a unique maximal definably connected definable subset of G containing the identity eG of G, which we call the definable identity component of G, and we denoted it by G0 . Thus, G is definably connected if and only if G = G0 , or, equivalently by [22], if G has no proper definable subgroup of finite index. We say that G is definably compact if every definable path γ : (0, 1) → G has limits points in G (where the limits are taken with respect to the t-topology on G). There is a close relation between groups definable in a field F and F -algebraic groups. Given an F -algebraic group H, the group of F -points H(F ) is a definable group in F . When F is an algebraically closed field, every definable group in F is F -definably isomorphic, as an F -definable group, to some F algebraic group ([6]). However, when F is real closed, there are semialgebraic groups over F that are not F -definably isomorphic to H(F ) for any F -alg
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