Geodesics with multiple self-intersections and symmetries on Riemann surfaces. (English) Zbl 0618.57001
Low dimensional topology and Kleinian groups, Symp. Warwick and Durham 1984, Lond. Math. Soc. Lect. Note Ser. 112, 3-11 (1986).
[For the entire collection see Zbl 0604.00015.]
Using a lemma of Nielsen, examples of closed geodesics on Riemann surfaces, resp. surfaces M of constant negative curvature, are constructed, which pass through a point v on the surface three or more times. These Riemann surfaces come from Fuchsian groups which admit as fundamental domain a rotationally symmetric polygon centred at the origin 0 in the Poincaré disk, with v as the image in M of 0. Using a group- theoretic analogue of Nielsen’s lemma the authors describe the free homotopy classes of the geodesics in question as words in \(\pi_ 1(M)\). It is also shown that the words corresponding to geodesics passing through the image in M of a vertex of the polygon contain ”half the defining relator” the same number of times as they pass through the image, and in addition satisfy certain symmetries.
Using a lemma of Nielsen, examples of closed geodesics on Riemann surfaces, resp. surfaces M of constant negative curvature, are constructed, which pass through a point v on the surface three or more times. These Riemann surfaces come from Fuchsian groups which admit as fundamental domain a rotationally symmetric polygon centred at the origin 0 in the Poincaré disk, with v as the image in M of 0. Using a group- theoretic analogue of Nielsen’s lemma the authors describe the free homotopy classes of the geodesics in question as words in \(\pi_ 1(M)\). It is also shown that the words corresponding to geodesics passing through the image in M of a vertex of the polygon contain ”half the defining relator” the same number of times as they pass through the image, and in addition satisfy certain symmetries.
Reviewer: B.Zimmermann
MSC:
| 57M05 | Fundamental group, presentations, free differential calculus |
| 30F10 | Compact Riemann surfaces and uniformization |
| 53C22 | Geodesics in global differential geometry |
| 57N05 | Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010) |
| 30F35 | Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) |
