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``We characterize discrete groups \(\Gamma\subset\text{GL}(n,{\mathbf R})\) which act properly discontinuously on the homogeneous space \(\text{GL}(m,{\mathbf R})\setminus\text{GL}(n,{\mathbf R})\). \((\dots)\) We now briefly outline the contents of the paper. In Section 2 we show that \(\text{GL}(m,\mathbf R) \backslash\text{GL} (n,\mathbf R)\) is essentially the Grassmannian \(G_{mn}\times M^0_{(n-m)n}\) -- the set of all \((n-m)\times n\) matrices of rank \(n-m\). We then give necessary and sufficient conditions for a discrete group \(\Gamma\subset \text{GL}(n,\mathbf R)\) to act properly discontinuously (from the right) on \(M_{kn}^0\). In Section 3 we characterize discrete groups \(\Gamma\subset\text{GL} (n,\mathbf R)\) which act properly discontinuously on \(G(m,\mathbf R)\backslash G(n,\mathbf R)\). Section 4 characterizes subgroups \(\Gamma\subset \text{GL } (n,\mathbf R)\), considered as groups of continuous transformations of a certain space of matrices \(S_{kn}\), which have an invariant probability measure. In the last section we show that if \(M=\text{GL} (m,\mathbf R)\setminus \text{GL}(n,\mathbf R)/ \Gamma\) is a compact manifold then \(\Gamma\), considered as a group of continuous transformations of \(G_{mn} \times S_{(n-m)m} \), does not have an invariant probability measure on \(G_{mn} \times S_{ (n-m)m }\)''.