Universal proper G-spaces

It is proved that: (1) every Lie group G can act properly (in sense of Palais) on each infinite-dimensional Hilbert space l2(τ) of a given weight τ such that (G,l2(τ)) becomes a universal G-space for all metrizable proper G-spaces admitting an invariant metric and having weight ⩽τ; (2) every Lie group G can act properly on Rτ⧹{0} such that (G,Rτ⧹{0}) becomes a universal G-space for all Tychonoff proper G-spaces of weight ⩽τ; (3) there is a dispersive dynamical system on l2, universal for all separable, metrizable, dispersive dynamical systems having a regular orbit space. Other universal proper G-spaces are constructed. As a corollary a shorter proof of Palaisʹ invariant metric existence theorem is obtained. The metric cones con(G/H), with H⊂G a compact subgroup, are the main building blocs in our approach.