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Optics in Croke-Kleiner spaces

Ancel F. D., Wilson J. M.

Bulletin of the Polish Academy of Sciences. Mathematics

2010

Vol. 58, no 2

147-165

We explore the interior geometry of the CAT(O) spaces {Xα : 0 < α ≤ π/2}, constructed by Croke and Kleiner [Topology 39 (2000)]. In particular, we describe a diffraction effect experienced by the family of geodesic rays that emanate from a basepoint and pass through a certain singular point called a triple point, and we describe the shadow this family casts on the boundary. This diffraction effect is codified in the Transformation Rules stated in Section 3 of this paper. The Transformation Rules have various applications. The earliest of these, described in Section 4, establishes a topological invariant of the boundaries of all the Xα's for which α lies in the interval [π/2(n + 1), π/2n), where n is a positive integer. Since the invariant changes when n changes, it provides a partition of the topological types of the boundaries of Croke-Kleiner spaces into a countable infinity of distinct classes. This countably infinite partition extends the original result of Croke and Kleiner which partitioned the topological types of the Croke-Kleiner boundaries into two distinct classes. After this countably infinite partition was proved, a finer partition of the topological types of the Croke-Kleiner boundaries into uncountably many distinct classes was established by the second author [J. Group Theory 8 (2005)], together with other applications of the Transformation Rules.