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Bounded Isometries and Homogeneous Quotients

In this paper we give an explicit description of the bounded displacement isometries of a class of spaces that includes the Riemannian nilmanifolds. The class of spaces consists of metric spaces (and thus includes Finsler manifolds) on which an exponential solvable Lie group acts transitively by isometries. The bounded isometries are proved to be of constant displacement. Their characterization gives further evidence for the author's 1962 conjecture on homogeneous Riemannian quotient manifolds. That conjecture suggests that if $Γ\backslash M$ is a Riemannian quotient of a connected simply connected homogeneous Riemannian manifold $M$, then $Γ\backslash M$ is homogeneous if and only if each isometry $γ\in Γ$ is of constant displacement. Our description of bounded isometries gives an alternative proof of an old result of J. Tits on bounded automorphisms of semisimple Lie groups. The topic of constant displacement isometries has an interesting history, starting with Clifford's use of quaternions in non--euclidean geometry, and we sketch that in a historical note.