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2010 | 58 | 2 | 147-165
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Optics in Croke-Kleiner Spaces

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We explore the interior geometry of the CAT(0) spaces ${X_{α}: 0 < α ≤ π/2}$, 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 $X_{α}$'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.
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  • Department of Mathematics, University of Wisconsin at Milwaukee, PO Box 413, Milwaukee, WI 53211, U.S.A.
  • Department of Mathematical Sciences, SUNY Fredonia, Fredonia, NY 14063, U.S.A.
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bwmeta1.element.bwnjournal-article-doi-10_4064-ba58-2-6
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