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

2008/11/01 by Ehud Hrushovski, Hrushovski, Ehud, Ya’acov Peterzil +3
Mathematics · #03C64 #22E15 #Advanced Topology and Set Theory #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.0811.0089

openalex publication_date 2008/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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, and (ii) the group (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 the full compact domination conjecture. 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, from which we can deduce the semialgebraicity of finite covers of Lie groups such as SL(2,R).

Citations

• On groups and fields definable in o-minimal structures

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