Universal G-spaces for proper actions of locally compact groups

We establish the existence of universal G-spaces for proper actions of locally compact groups on Tychonoff spaces. A typical result sounds as follows: for each infinite cardinal number τ every locally compact, non-compact, σ-compact group G of weight w ( G ) ⩽ τ , can act properly on R τ ∖ { 0 } such that R τ ∖ { 0 } contains a G-homeomorphic copy of every Tychonoff proper G-space of weight ⩽τ. The metric cones Cone ( G / H ) with H ⊂ G a compact subgroup such that G / H is a manifold, are the main building blocks in our approach. As a byproduct we prove that the cardinality of the set of all conjugacy classes of such subgroups H ⊂ G does not exceed the weight of G.