A characterization of G-ANR and G-AR spaces for proper actions of Lie groups

Let G be a matrix Lie group. We prove that a proper G-space X of finite structure, which is metrizable by a G-invariant metric, is a G-ANR (resp., a G-AR) iff for any compact subgroup H⊂G the H-fixed point set XH is an ANR (resp., an AR). An equivariant embedding result for proper G-spaces of finite structure is also obtained.