Topological equivalence of foliations of homogeneous spaces

Abstract:

For $i = 1,2$, let ${\Gamma _i}$ be a lattice in a connected Lie group ${G_i}$, and let ${X_i}$ be a connected Lie subgroup of ${G_i}$. The double cosets ${\Gamma _i}g{X_i}$ provide a foliation ${\mathcal {F}_i}$ of the homogeneous space ${\Gamma _i}\backslash {G_i}$. Assume that ${X_1}$ and ${X_2}$ are unimodular and that ${\mathcal {F}_1}$ has a dense leaf. If ${G_1}$ and ${G_2}$ are semisimple groups to which the Mostow Rigidity Theorem applies, or are simply connected nilpotent groups (or are certain more general solvable groups), we use an idea of D. Benardete to show that any topological equivalence of ${\mathcal {F}_1}$ and ${\mathcal {F}_2}$ must be the composition of two very elementary maps: an affine map and a map that takes each leaf to itself.