Geodesic
Spaces of closed subgroups of a connected Lie group
Glasgow mathematical journal, Tome 14 (1973) no. 1, pp. 77-79

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In a sequence of two papers which appeared in 1968 and 1969 Herbert Abels [1, 2] has developed, from a method originated by Gerstenhaber [6], a means for extending the study of properly discontinuous groups of transformations to that of proper transformation groups in general. We recall that, if G is a Hausdorff locally compact group of transformations of a locally compact space X, then the action of Gis proper when, for any two compact subsets K and L, the subset G(K, L) = {g ɛ G: gL∩K # 0} of G is compact (see [3], p. 55). In what follows all groups and spaces will be Hausdorff and locally compact. If H is a closed subgroup of G, then it is clear that the property just defined is possessed by the action of H as a group of left translations of G. 
Oler, N. Spaces of closed subgroups of a connected Lie group. Glasgow mathematical journal, Tome 14 (1973) no. 1, pp. 77-79. doi: 10.1017/S0017089500001762
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[1] 1.Abels, H., Über die Erzengung von eigentlichen Transformationsgruppen, Math. Zeit. 103 (1968), 333–357. Google Scholar | DOI

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