Connected components of definable groups and o-minimality. I
type: Item
instance of: scholarly article
title: Connected components of definable groups and o-minimality. I
zbMATH Open document ID: 1255.03041
author: Annalisa Conversano
author: Anand Pillay
published in: Advances in Mathematics
publication date: 2012-09-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/1101.5705
reviewed by: Eric Jaligot
Mathematics Subject Classification ID: 03C64
zbMATH DE Number: 6084075
zbMATH Keywords: definable group
zbMATH Keywords: o-minimality
zbMATH Keywords: amenability
zbMATH Keywords: bounded orbit
MaRDI profile type: Publication
OpenAlex ID: W2158457721
arXiv ID: 1101.5705
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DOI: 10.1016/J.AIM.2012.05.022
review text: The authors consider the ``connected components'' \(G^{00}\) and \(G^{000}\) of groups \(G\) definable in o-minimal expansions of real closed fields, or structures closely related to o-minimal structures. Most of the results in this paper follow from structural results on definable groups from the first author's doctoral thesis. These structural results are reviewed in the second section, with notable decompositions involving the various simple factors, compact or not, of the semisimple part of the group. Everything starts in Proposition 2.1 with a proof of the existence of a maximal normal definable torsion-free (and automatically solvable) subgroup, which essentially relies on a lemma on lifting of torsion. In the paper that lemma is proved by invoking very specific machinery on Euler characteristic in o-minimal structures, but it is worth mentioning that it is valid in any group with the descending chain condition on definable subgroups (by arguments similar to the usual proof in the finite Morley rank case).NEWLINENEWLINEIn the third section the authors prove that if \(G\) is a saturated model of the universal cover of \(SL_2(R)\) (interpretable in the two-sorted structure consisting of the field \(\mathbb R\) of reals and the infinite cyclic group), then \(G^{000}\) is a proper subgroup of \(G^{00}=G\). They then give a group definable in the reals and with a similar property (and expect that this property is indeed typical). These groups also provide new ``natural'' examples of theories where the group of Lascar strong automorphisms is properly contained in the group of Kim-Pillay strong automorphisms. In the fourth section the authors raise questions about definable amenability and the nature of types with bounded orbits in the o-minimal and, more generally, NIP environment. In particular they obtain that for a group definable in an o-minimal expansion of a real closed field, the existence of a type with a bounded orbit is equivalent to the definable amenabil…
description: scientific article; zbMATH DE number 6084075