Optics in Croke–Kleiner Spaces

Fredric D. Ancel ¹   ; Julia M. Wilson ²

¹ Department of Mathematics University of Wisconsin at Milwaukee PO Box 413 Milwaukee, WI 53211, U.S.A.
² Department of Mathematical Sciences SUNY Fredonia Fredonia, NY 14063, U.S.A.

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 58 (2010) no. 2, pp. 147-165

Résumé

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.

Détail

DOI : 10.4064/ba58-2-6

Keywords: explore interior geometry cat spaces alpha alpha leq constructed croke kleiner topology particular describe diffraction effect experienced family geodesic rays emanate basepoint pass through certain singular point called triple point describe shadow family casts boundary diffraction effect codified transformation rules stated section paper transformation rules have various applications earliest these described section establishes topological invariant boundaries alpha which alpha lies interval where positive integer since invariant changes changes provides partition topological types boundaries croke kleiner spaces countable infinity distinct classes countably infinite partition extends original result croke kleiner which partitioned topological types croke kleiner boundaries distinct classes after countably infinite partition proved finer partition topological types croke kleiner boundaries uncountably many distinct classes established second author group theory together other applications transformation rules