Topological groups, μ-types and their stabilizers

Ya'Acov Peterzil; Sergei Starchenko

doi:10.4171/JEMS/733

Topological groups, μ-types and their stabilizers

Ya'Acov Peterzil, Sergei Starchenko

Department of Mathematics

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

Abstract

We consider an arbitrary topological group G definable in a structure M, such that some basis for the topology of G consists of sets definable in M. To each such group G we associate a compact G-space of partial types, S^μ_G.(M) = {pμ : p ϵ S_G.(M)}, which is the quotient of the usual type space S_G.(M) by the relation of two types being "infinitesimally close to each other". In the o-minimal setting, if p is a definable type then it has a corresponding definable subgroup Stab^μ.(p), which is the stabilizer of pμ. This group is nontrivial when p is unbounded; in fact, it is a torsion-free solvable group. Along the way, we analyze the general construction of S^μ_G.(M) and its connection to the Samuel compactification of topological groups.

Original language	English
Pages (from-to)	2965-2995
Number of pages	31
Journal	Journal of the European Mathematical Society
Volume	19
Issue number	10
DOIs	https://doi.org/10.4171/JEMS/733
State	Published - 2017

Bibliographical note

Funding Information:
The authors thank the US-Israel Binational Science Foundation for its support, and also thank the Mathematical Science Research Institute at Berkeley for its hospitality during Spring 2014. The second author thanks the National Science Foundation for support.

Keywords

Compactification
Definable groups
O-minimality

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.4171/JEMS/733

Cite this

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title = "Topological groups, μ-types and their stabilizers",

abstract = "We consider an arbitrary topological group G definable in a structure M, such that some basis for the topology of G consists of sets definable in M. To each such group G we associate a compact G-space of partial types, SμG.(M) = {pμ : p ϵ SG.(M)}, which is the quotient of the usual type space SG.(M) by the relation of two types being {"}infinitesimally close to each other{"}. In the o-minimal setting, if p is a definable type then it has a corresponding definable subgroup Stabμ.(p), which is the stabilizer of pμ. This group is nontrivial when p is unbounded; in fact, it is a torsion-free solvable group. Along the way, we analyze the general construction of SμG.(M) and its connection to the Samuel compactification of topological groups.",

keywords = "Compactification, Definable groups, O-minimality",

author = "Ya'Acov Peterzil and Sergei Starchenko",

note = "Funding Information: The authors thank the US-Israel Binational Science Foundation for its support, and also thank the Mathematical Science Research Institute at Berkeley for its hospitality during Spring 2014. The second author thanks the National Science Foundation for support.",

year = "2017",

doi = "10.4171/JEMS/733",

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T1 - Topological groups, μ-types and their stabilizers

AU - Peterzil, Ya'Acov

AU - Starchenko, Sergei

N1 - Funding Information: The authors thank the US-Israel Binational Science Foundation for its support, and also thank the Mathematical Science Research Institute at Berkeley for its hospitality during Spring 2014. The second author thanks the National Science Foundation for support.

PY - 2017

Y1 - 2017

N2 - We consider an arbitrary topological group G definable in a structure M, such that some basis for the topology of G consists of sets definable in M. To each such group G we associate a compact G-space of partial types, SμG.(M) = {pμ : p ϵ SG.(M)}, which is the quotient of the usual type space SG.(M) by the relation of two types being "infinitesimally close to each other". In the o-minimal setting, if p is a definable type then it has a corresponding definable subgroup Stabμ.(p), which is the stabilizer of pμ. This group is nontrivial when p is unbounded; in fact, it is a torsion-free solvable group. Along the way, we analyze the general construction of SμG.(M) and its connection to the Samuel compactification of topological groups.

AB - We consider an arbitrary topological group G definable in a structure M, such that some basis for the topology of G consists of sets definable in M. To each such group G we associate a compact G-space of partial types, SμG.(M) = {pμ : p ϵ SG.(M)}, which is the quotient of the usual type space SG.(M) by the relation of two types being "infinitesimally close to each other". In the o-minimal setting, if p is a definable type then it has a corresponding definable subgroup Stabμ.(p), which is the stabilizer of pμ. This group is nontrivial when p is unbounded; in fact, it is a torsion-free solvable group. Along the way, we analyze the general construction of SμG.(M) and its connection to the Samuel compactification of topological groups.

KW - Compactification

KW - Definable groups

KW - O-minimality

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DO - 10.4171/JEMS/733

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JF - Journal of the European Mathematical Society

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ER -

Topological groups, μ-types and their stabilizers

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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