Proper actions of locally compact groups on equivariant absolute extensors

• Autorzy: Sergey Antonyan
• Języki publikacji: EN

Abstrakty

EN

Let G be a locally compact Hausdorff group. We study equivariant absolute (neighborhood) extensors (G-AE's and G-ANE's) in the category G-ℳ of all proper G-spaces that are metrizable by a G-invariant metric. We first solve the linearization problem for proper group actions by proving that each X ∈ G-ℳ admits an equivariant embedding in a Banach G-space L such that L∖{0} is a proper G-space and L∖{0} ∈ G-AE. This implies that in G-ℳ the notions of G-A(N)E and G-A(N)R coincide. Our embedding result is applied to prove that if a G-space X is a G-ANE (resp., a G-AE) such that all the orbits in X are metrizable, then the orbit space X/G is an ANE (resp., an AE if, in addition, G is almost connected). Furthermore, we prove that if X ∈ G-ℳ then for any closed embedding X/G ↪ B in a metrizable space B, there exists a closed G-embedding X ↪ Z (a lifting) in a G-space Z ∈ G-ℳ such that Z/G is a neighborhood of X/G (resp., Z/G = B whenever G is almost connected). If a proper G-space X has metrizable orbits and a metrizable orbit space then it is metrizable (by a G-invariant metric).

Kategorie tematyczne

• 57S99: None of the above, but in this section
• 54C55: Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)
• 54H15: Transformation groups and semigroups

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2009
• Tom: 205
• Numer: 2
• Strony: 117-145

Daty

wydano

2009

Twórcy

autor

Sergey Antonyan

• Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, 04510 México D.F., Mexico

Identyfikatory

DOI

10.4064/fm205-2-3