Optics in Croke–Kleiner Spaces
Volume 58 / 2010
Abstract
We explore the interior geometry of the CAT(0) spaces $\{ X_{\alpha} : 0 < \alpha \leq {\pi}/{2} \}$, 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 $X_{\alpha}$'s for which $\alpha$ lies in the interval $[{\pi}/{2(n+1)},{\pi}/{2n})$, 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.
