There are reductive homogeneous spaces admitting proper actions of cocompact lattices of O(n,1) for n=2,3,4 while admitting no proper action of a non-compact semisimple subgroup.
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Exotic proper actions on homogeneous spaces via convex cocompact representations
There are reductive homogeneous spaces admitting proper actions of cocompact lattices of O(n,1) for n=2,3,4 while admitting no proper action of a non-compact semisimple subgroup.