Optics in Croke-Kleiner Spaces

Fredric D. Ancel, and Julia M. Wilson. "Optics in Croke-Kleiner Spaces." Bulletin of the Polish Academy of Sciences. Mathematics 58.2 (2010): 147-165. <http://eudml.org/doc/281348>.

@article{FredricD2010,
abstract = {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.},
author = {Fredric D. Ancel, Julia M. Wilson},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {Croke-Kleiner space; group; boundary; topological types},
language = {eng},
number = {2},
pages = {147-165},
title = {Optics in Croke-Kleiner Spaces},
url = {http://eudml.org/doc/281348},
volume = {58},
year = {2010},
}

TY - JOUR
AU - Fredric D. Ancel
AU - Julia M. Wilson
TI - Optics in Croke-Kleiner Spaces
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2010
VL - 58
IS - 2
SP - 147
EP - 165
AB - 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.
LA - eng
KW - Croke-Kleiner space; group; boundary; topological types
UR - http://eudml.org/doc/281348
ER -