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

Ehud Hrushovski; Ya'acov Peterzil; Anand Pillay

doi:10.1016/j.jalgebra.2010.11.001

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

Ehud Hrushovski, Ya'acov Peterzil, Anand Pillay

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

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.

Original language	English
Pages (from-to)	71-106
Number of pages	36
Journal	Journal of Algebra
Volume	327
Issue number	1
DOIs	https://doi.org/10.1016/j.jalgebra.2010.11.001
State	Published - 1 Feb 2011

Bibliographical note

Funding Information:
✩ The first author thanks ISF grant 1048-078, and travel funds from an EPSRC grant. The third author thanks the European Commission (Marie Curie Excellence chair), the EPSRC (grant EP/F009712/1), as well as Université Paris-Sud 11. All authors thank the Marie Curie Modnet network for facilitating the collaboration. * Corresponding author. E-mail addresses: [email protected] (E. Hrushovski), [email protected] (Y. Peterzil), [email protected] (A. Pillay).

Keywords

Central extensions
Compact domination
Definably compact
O-minimality
Semialgebraic groups
Universal covers

ASJC Scopus subject areas

Algebra and Number Theory

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10.1016/j.jalgebra.2010.11.001

Cite this

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abstract = "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.",

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author = "Ehud Hrushovski and Ya'acov Peterzil and Anand Pillay",

note = "Funding Information: ✩ The first author thanks ISF grant 1048-078, and travel funds from an EPSRC grant. The third author thanks the European Commission (Marie Curie Excellence chair), the EPSRC (grant EP/F009712/1), as well as Universit{\'e} Paris-Sud 11. All authors thank the Marie Curie Modnet network for facilitating the collaboration. * Corresponding author. E-mail addresses: [email protected] (E. Hrushovski), [email protected] (Y. Peterzil), [email protected] (A. Pillay).",

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KW - Compact domination

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

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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