Abelian pro-countable groups and orbit equivalence relations

Maciej Malicki. "Abelian pro-countable groups and orbit equivalence relations." Fundamenta Mathematicae 233.1 (2016): 83-99. <http://eudml.org/doc/286522>.

@article{MaciejMalicki2016,
abstract = { We study a class of abelian groups that can be defined as Polish pro-countable groups, as non-archimedean groups with a compatible two-sided invariant metric or as quasi-countable groups, i.e., closed subdirect products of countable discrete groups, endowed with the product topology. We show that for every non-locally compact, abelian quasi-countable group G there exists a closed L ≤ G and a closed, non-locally compact K ≤ G/L which is a direct product of discrete countable groups. As an application we prove that for every abelian Polish group G of the form H/L, where H,L ≤ Iso(X) and X is a locally compact separable metric space (in particular, for every abelian, quasi-countable group G), the following holds: G is locally compact iff every continuous action of G on a Polish space Y induces an orbit equivalence relation that is reducible to an equivalence relation with countable classes. },
author = {Maciej Malicki},
journal = {Fundamenta Mathematicae},
keywords = {non-locally compact Polish groups; abelian groups; orbit equivalence relations; quasicountable groups; procountable groups},
language = {eng},
number = {1},
pages = {83-99},
title = {Abelian pro-countable groups and orbit equivalence relations},
url = {http://eudml.org/doc/286522},
volume = {233},
year = {2016},
}

TY - JOUR
AU - Maciej Malicki
TI - Abelian pro-countable groups and orbit equivalence relations
JO - Fundamenta Mathematicae
PY - 2016
VL - 233
IS - 1
SP - 83
EP - 99
AB - We study a class of abelian groups that can be defined as Polish pro-countable groups, as non-archimedean groups with a compatible two-sided invariant metric or as quasi-countable groups, i.e., closed subdirect products of countable discrete groups, endowed with the product topology. We show that for every non-locally compact, abelian quasi-countable group G there exists a closed L ≤ G and a closed, non-locally compact K ≤ G/L which is a direct product of discrete countable groups. As an application we prove that for every abelian Polish group G of the form H/L, where H,L ≤ Iso(X) and X is a locally compact separable metric space (in particular, for every abelian, quasi-countable group G), the following holds: G is locally compact iff every continuous action of G on a Polish space Y induces an orbit equivalence relation that is reducible to an equivalence relation with countable classes.
LA - eng
KW - non-locally compact Polish groups; abelian groups; orbit equivalence relations; quasicountable groups; procountable groups
UR - http://eudml.org/doc/286522
ER -