Minimal actions of homeomorphism groups

Yonatan Gutman. "Minimal actions of homeomorphism groups." Fundamenta Mathematicae 198.3 (2008): 191-215. <http://eudml.org/doc/283236>.

@article{YonatanGutman2008,
abstract = {Let X be a closed manifold of dimension 2 or higher or the Hilbert cube. Following Uspenskij one can consider the action of Homeo(X) equipped with the compact-open topology on $Φ ⊂ 2^\{2^\{X\}\}$, the space of maximal chains in $2^\{X\}$, equipped with the Vietoris topology. We show that if one restricts the action to M ⊂ Φ, the space of maximal chains of continua, then the action is minimal but not transitive. Thus one shows that the action of Homeo(X) on $U_\{Homeo(X)\}$, the universal minimal space of Homeo(X), is not transitive (improving a result of Uspenskij). Additionally for X as above with dim(X) ≥ 3 we characterize all the minimal subspaces of V(M), the space of closed subsets of M, and show that M is the only minimal subspace of Φ. For dim(X) ≥ 3, we also show that (M,Homeo(X)) is strongly proximal.},
author = {Yonatan Gutman},
journal = {Fundamenta Mathematicae},
keywords = {homeomorphism group; minimal action; Hilbert cube},
language = {eng},
number = {3},
pages = {191-215},
title = {Minimal actions of homeomorphism groups},
url = {http://eudml.org/doc/283236},
volume = {198},
year = {2008},
}

TY - JOUR
AU - Yonatan Gutman
TI - Minimal actions of homeomorphism groups
JO - Fundamenta Mathematicae
PY - 2008
VL - 198
IS - 3
SP - 191
EP - 215
AB - Let X be a closed manifold of dimension 2 or higher or the Hilbert cube. Following Uspenskij one can consider the action of Homeo(X) equipped with the compact-open topology on $Φ ⊂ 2^{2^{X}}$, the space of maximal chains in $2^{X}$, equipped with the Vietoris topology. We show that if one restricts the action to M ⊂ Φ, the space of maximal chains of continua, then the action is minimal but not transitive. Thus one shows that the action of Homeo(X) on $U_{Homeo(X)}$, the universal minimal space of Homeo(X), is not transitive (improving a result of Uspenskij). Additionally for X as above with dim(X) ≥ 3 we characterize all the minimal subspaces of V(M), the space of closed subsets of M, and show that M is the only minimal subspace of Φ. For dim(X) ≥ 3, we also show that (M,Homeo(X)) is strongly proximal.
LA - eng
KW - homeomorphism group; minimal action; Hilbert cube
UR - http://eudml.org/doc/283236
ER -