A descending chain condition for groups definable in o-minimal structures

Alessandro Berarducci; Margarita Otero; Yaa'cov Peterzil; Anand Pillay

doi:10.1016/j.apal.2005.01.002

A descending chain condition for groups definable in o-minimal structures

Alessandro Berarducci, Margarita Otero, Yaa'cov Peterzil, Anand Pillay

Department of Mathematics

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

Abstract

We prove that if G is a group definable in a saturated o-minimal structure, then G has no infinite descending chain of type-definable subgroups of bounded index. Equivalently, G has a smallest (necessarily normal) type-definable subgroup G⁰⁰ of bounded index and G/G⁰⁰ equipped with the "logic topology" is a compact Lie group. These results give partial answers to some conjectures of the fourth author.

Original language	English
Pages (from-to)	303-313
Number of pages	11
Journal	Annals of Pure and Applied Logic
Volume	134
Issue number	2-3
DOIs	https://doi.org/10.1016/j.apal.2005.01.002
State	Published - Jul 2005

Bibliographical note

Funding Information:
The first two authors thank Dikran Dikranjan for useful information on compact groups. M. Otero was partially supported by BFM-2002-04797. Y. Peterzil was partially supported by funds from the NSF Focused Research Grant DMS-0100979. A. Pillay was partially supported by NSF grants DMS-0300639 and the FRG DMS-0100979.

Keywords

Compact groups
Descending chain conditions
Type-definable subgroups
o-minimality

ASJC Scopus subject areas

Logic

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10.1016/j.apal.2005.01.002

Cite this

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abstract = "We prove that if G is a group definable in a saturated o-minimal structure, then G has no infinite descending chain of type-definable subgroups of bounded index. Equivalently, G has a smallest (necessarily normal) type-definable subgroup G00 of bounded index and G/G00 equipped with the {"}logic topology{"} is a compact Lie group. These results give partial answers to some conjectures of the fourth author.",

keywords = "Compact groups, Descending chain conditions, Type-definable subgroups, o-minimality",

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AU - Otero, Margarita

AU - Peterzil, Yaa'cov

AU - Pillay, Anand

N1 - Funding Information: The first two authors thank Dikran Dikranjan for useful information on compact groups. M. Otero was partially supported by BFM-2002-04797. Y. Peterzil was partially supported by funds from the NSF Focused Research Grant DMS-0100979. A. Pillay was partially supported by NSF grants DMS-0300639 and the FRG DMS-0100979.

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N2 - We prove that if G is a group definable in a saturated o-minimal structure, then G has no infinite descending chain of type-definable subgroups of bounded index. Equivalently, G has a smallest (necessarily normal) type-definable subgroup G00 of bounded index and G/G00 equipped with the "logic topology" is a compact Lie group. These results give partial answers to some conjectures of the fourth author.

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KW - Compact groups

KW - Descending chain conditions

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ER -

A descending chain condition for groups definable in o-minimal structures

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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