Topological dynamics for groups definable in real closed field
type: Item
instance of: scholarly article
title: Topological dynamics for groups definable in real closed field
zbMATH Open document ID: 1331.03029
author: Ningyuan Yao
author: Dong-Yang Long
published in: Annals of Pure and Applied Logic
publication date: 2015-01-23T00:00:00Z
review text: This paper gives an extension of results by \textit{G. Jagiella} [Math. Log. Q. 61, No. 1--2, 45--55 (2015; Zbl 1341.03049)]. The main theorem of this paper says that for a group \(G\) definable over an o-minimal expansion of the reals and having a decomposition \(G=HK\), \(K\) definably compact and \(H\) torsion free, and for an elementary extension \(M\) of \(\mathbb{R}\), the Ellis group of the Ellis semigroup \(S_G(M^{\mathrm{ext}})\) of \(S_G(M)\), where \(S_G(M)\) is the space of types of \(G\) over \(M\), is algebraically isomorphic to \((N_G(H) \cap K) (\mathbb{R})\). Here, \(S_G(M^{\mathrm{ext}})\) is the space of external types of \(G\) over \(M\). This leads the authors to extend a counter-example to a conjecture of Newelski saying that for a theory with NIP, the Ellis groups of the Ellis semigroup (the universal definable \(G(M)\)-flow) \(S_G(M^{\mathrm{ext}})\) of the \(G(M)\)-flow \(S_G(M)\) are isomorphic to \(G/G^{00}\). The counter-example is \(\mathrm{SL}(2,M)\) with \(M\) an elementary extension of the reals, not just the reals. The case \(M\neq \mathbb{R}\) is shown to be different from the case \(M=\mathbb{R}\).
reviewed by: Artur Piękosz
Mathematics Subject Classification ID: 03C64
Mathematics Subject Classification ID: 37B05
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 6389865
zbMATH Keywords: o-minimality
zbMATH Keywords: topological dynamics
zbMATH Keywords: compact-torsion-free decomposition
zbMATH Keywords: minimal flows
zbMATH Keywords: Ellis group
MaRDI profile type: Publication
full work available at URL: j.apal.2014.10.004
OpenAlex ID: W2058237250
cites work: Q4040097
cites work: External definability and groups in NIP theories
cites work: Q5589084
cites work: On compactifications and the topological dynamics of definable groups
cites work: Some model theory of SL(2,R)
cites work: Groups, measures, and the NIP
cites work: Definable topological dynamics and real Lie groups
cites work: Definable Sets in Ordered Structures. II
cites work: Definable types in -minimal theories
cites work: Topological dynamics of definable group actions
cites work: Model theoretic aspects of the Ellis semigroup
cites work: Q3130253
cites work: Topological dynamics and definable groups
cites work: A course in model theory. An introduction to contemporary mathematical logic. Transl. from the French by Moses Klein
cites work: Dependent first order theories, continued
DOI: 10.1016/J.APAL.2014.10.004
Recommended article: Model theoretic aspects of the Ellis semigroup
Recommended article: Definable topological dynamics and real Lie groups
Recommended article: Some model theory of SL(2,R)
Recommended article: Topological dynamics and definable groups
Recommended article: External definability and groups in NIP theories
Recommended article: Topological dynamics and NIP fields
Recommended article: Boundedness and absoluteness of some dynamical invariants in model theory
Recommended article: THE ELLIS GROUP CONJECTURE AND VARIANTS OF DEFINABLE AMENABILITY
Recommended article: Connected components of definable groups and o-minimality. I
Recommended article: On compactifications and the topological dynamics of definable groups
description: scientific article; zbMATH DE number 6389865