| It is a famous result due to Borel[29] and Harish-Chandra[29] that homogeneousspace G/H admits a uniform lattice in 1962, i.e. if H is compact , there is a discretesubgroupΓin G acting proper discontinuously and freely on G/H so thatΓ\G/H iscompact.But unless H is compact ,the action of a discrete group on G/H is not automaticallyproper discontinuously . In fact it sometimes occurs that only finite subgroup in Gcan act proper discontinuously on G/H. Calabi[2] and Markus[2] first found SO(n +1,1)/SO(n,1) is such a case in 1962. Now some su?cient conditions on these Calabi-Markus phenomena have been obtained in a general case .To study a proper discontinuously action on a reductive homogeneous space G/H,wewill take the following approach: find a reductive subgroup G acting on G/H properly, so that any discrete subgroupΓof G acts automatically properly discontinuously onG/H. This approach was first partially carried out by Kulkarni. The main results of thispaper is that letting G = SL(n,R),H = SL(m,R),L = SL(2,R), I mainly check that Lacting on G/H is proper . |
