## 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.