Simple closed geodesics in hyperbolic 3-manifolds. (English) Zbl 0955.53025

Adams, Colin; Hass, Joel; Scott, Peter

Bull. Lond. Math. Soc. 31, No. 1, 81-86 (1999).

It is known that among the orientable, finite area, complete hyperbolic 2-manifolds, the thrice-punctured sphere is the only example that contains no simple closed geodesics. The authors investigate the question which orientable hyperbolic 3-manifolds do and do not contain simple closed geodesics. Based on a geometric characterization of non-screw isometries of hyperbolic space \(H^3\), the following result is proven: Let \(M\) be an oriented hyperbolic 3-manifold. Then exactly one of the following three cases occurs. (1) There exists a simple closed geodesic in \(M\). (2) \(M\) is the quotient of \(H^3\) by a Fuchsian group corresponding to the thrice-punctured sphere. (3) The fundamental group of \(M\) is elementary with zero or one limit point.

Reviewer: Ruth Kellerhals (Talence)

MSC:

53C22	Geodesics in global differential geometry
57M50	General geometric structures on low-dimensional manifolds

Keywords:

hyperbolic 3-manifold; closed geodesic; screw motion; thrice-punctured sphere