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Some properties of horocycles on Damek-Ricci spaces. (English) Zbl 1156.43004
Damek-Ricci spaces form a large class of solvable Lie groups equipped with a suitable Riemannian metric. They are named after E. Damek and F. Ricci, who noted that they are harmonic spaces. This class includes all rank-one symmetric spaces of the noncompact type. However, many Damek-Ricci spaces are not symmetric, i.e., the geodesic inversion around the origin is not an isometry, so nonsymmetric Damek-Ricci spaces provide counterexamples to the Lichnerowicz conjecture.
Several authors have studied analysis on these spaces in order to understand the differences between symmetric and nonsymmetric harmonic spaces. Although radial analysis is essentially the same in both cases, the study of nonradial analysis is often much more complicated in the nonsymmetric case. This is due to the lack of a group acting transitively by isometries on geodesic spheres.
In this paper the authors study some basic properties of horocycles on Damek-Ricci spaces. Horocycles generalize the notion of hyperplane in $$\mathbb R^n$$ and they are defined as level sets of the Busemann function. One can write the Busemann function in terms of Poisson kernels. As a consequence, horocycles can also be viewed as level sets of the Poisson kernel.
The main result of the paper is that a Damek-Ricci space is symmetric iff the geodesic inversion preserves the set of horocycles. In the proof the authors use the fact that symmetric spaces are characterized among all Damek-Ricci spaces by a purely algebraic condition, called the $$\jmath^2$$-condition.

##### MSC:
 43A80 Analysis on other specific Lie groups 37-06 Proceedings, conferences, collections, etc. pertaining to dynamical systems and ergodic theory 22E25 Nilpotent and solvable Lie groups 37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
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