## 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.

### MSC:

 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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### References:

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