Optics in Croke-Kleiner spaces

• Autorzy: Ancel F. D. , Wilson J. M.
• Języki publikacji: EN

Abstrakty

EN

We explore the interior geometry of the CAT(O) spaces {Xα : 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 Xα'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.

Słowa kluczowe

EN

Croke-Kleiner group; geometric group theory

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2010
• Tom: Vol. 58, no 2
• Strony: 147--165
• Opis fizyczny: Bibliogr. 7 poz.

Twórcy

autor

Ancel F. D.

autor

Wilson J. M.

• Department of Mathematics University of Wisconsin at Milwaukee, PO Box 413, Milwaukee, WI 53211, U.S.A.,

Bibliografia

• [1] F. Ancel, C. Guilbault, and J. Wilson, The Croke-Kleiner boundaries are cell-like equivalent, preprint.
• [2] M. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer, Berlin, 1999.
• [3] P. Bowers and K. Ruane, Boundaries of nonpositively curved groups of the form G x Zn, Glasgow Math. J. 38 (1996), 177-189.
• [4] C. Croke and B. Kleiner, Spaces with nonpositive curvature and their ideal boundaries, Topology 39 (2000), 549-556.
• [5] M. W. Davis, Nonpositive curvature and reflection groups, in: Handbook of Geometric Topology, R. J. Daverman and R. B. Sher (eds.), Elsevier, Amsterdam, 2002, 373-422.
• [6] E. Ghys and P. de la Harpe (eds.), Sur les Groupes Hyperboliques d’apres Mikhael Gromov, Birkhäuser, Boston, 1990.
• [7] J. M. Wilson, A CAT(O) group with uncountably many distinct boundaries, J. Group Theory 8 (2005), 229-238.