Hyperbolic Structures on the Configuration Space of Six Points in the Projective Line

✍ Scribed by Bernard Morin; Haruko Nishi

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 379 KB
• Volume: 150
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1999.1840

✦ Synopsis

The oriented configuration space X + 6 of six points on the real projective line is a noncompact three-dimensional manifold which admits a unique complete hyperbolic structure of finite volume with ten cusps. On the other hand, it decomposes naturally into 120 cells each of which can be interpreted as the set of equiangular hexagons with unit area. Similar hyperbolic structures can be obtained by considering nonequiangular hexagons so that the standard hyperbolic structure on X + 6 is at the center of a five parameter family of hyperbolic structures of finite volume. This paper contributes to investigations of the properties of this family. In particular, we exhibit two real analytic maps from the set of prescribed angles of hexagons into R 10 whose components are the traces of the monodromies at the ten cusps. We show that this map has maximal rank 5 at the center.