Existence of compact quotients of homogeneous spaces, measurably proper actions, and decay of matrix coefficients. (English) Zbl 0892.22009

Let \(G\) be a Lie group and \(H\) a closed subgroup of \(G\). If \(H\) is noncompact it is not known when \(G/H\) has a compact quotient (even for semisimple \(G\)). There are several criteria, however, to prove that a given \(G/H\) has no compact quotients. These criteria are based on considerations from topology, ergodic theory and the theory of linear groups. In this paper, the author gives a new criterion based on the study of the restriction to \(H\) of matrix coefficients of unitary representations of \(G\). Using this criterion, new examples of homogeneous spaces \(G/H\) without compact quotients can be generated.


22E40 Discrete subgroups of Lie groups
22E46 Semisimple Lie groups and their representations
53C30 Differential geometry of homogeneous manifolds
57S30 Discontinuous groups of transformations
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