Universal G-spaces for proper free actions

For a Lie group G, we prove the existence of a universal G-space in the class of all paracompact (respectively, metrizable, and separable metrizable) free proper G-spaces which have a paracompact orbit space. Our approach is based on Milnor's construction E G . We show that E G and the universal free G-spaces in question are G-AE's. Besides, we show how to extend these results to the case of arbitrary metrizable, locally compact, almost connected group actions.