Schottky group

In mathematics, a Schottky group is a special sort of Kleinian group, first studied by Friedrich Schottky (1877).

Definition

Fix some point p on the Riemann sphere. Each Jordan curve not passing through p divides the Riemann sphere into two pieces, and we call the piece containing p the "exterior" of the curve, and the other piece its "interior". Suppose there are 2g disjoint Jordan curves A₁, B₁,..., A_g, B_g in the Riemann sphere with disjoint interiors. If there are Moebius transformations T_i taking the outside of A_i onto the inside of B_i, then the group generated by these transformations is a Kleinian group. A Schottky group is any Kleinian group that can be constructed like this.

Properties

Schottky groups are finitely generated free groups such that all non-trivial elements are loxodromic. Conversely Maskit (1967) showed that any finitely generated free Kleininan group such that all non-trivial elements are loxodromic is a Schottky group.

A fundamental domain for the action of a Schottky group G on its regular points Ω(G) in the Riemann sphere is given by the exterior of the Jordan curves defining it. The corresponding quotient space Ω(G)/G is given by joining up the Jordan curves in pairs, so is a compact Riemann surface of genus g. This is the boundary of the 3-manifold given by taking the quotient (H∪Ω(G))/G of 3-dimensional hyperbolic H space plus the regular set Ω(G) by the Schottky group G, which is a handlebody of genus g. Conversely any compact Riemann surface of genus g can be obtained from some Schottky group of genus g.

Classical and non-classical Schottky groups

A Schottky group is called classical if all the disjoint Jordan curves corresponding to some set of generators can be chosen to be circles. Marden (1974, 1977) gave an indirect and non-constructive proof of the existence of non-classical Schottky groups, and Yamamoto (1991) gave an explicit example of one. It has been shown by Peter Doyle (1988) that all finitely generated classical Schottky groups have limit sets of Hausdorff dimension bounded above strictly by a universal constant less than 2.

Schottky space

Schottky space (of some genus g≥2) is the space of marked Schottky groups of genus g, in other words the space of sets of g elements of PSL₂(C) that generate a Schottky group, up to equivalence under Moebius transformations (Bers 1975). It is a complex manifold of complex dimension 3g−3. It contains classical Schottky space as the subset corresponding to classical Schottky groups.

Schottky space of genus g is not simply connected in general, but its universal covering space can be identified with Teichmüller space of compact genus g Riemann surfaces.

References

• Bers, Lipman (1975), "Automorphic forms for Schottky groups", Advances in Mathematics 16: 332–361, doi:10.1016/0001-8708(75)90117-6, ISSN 0001-8708, MR0377044 
• Chuckrow, Vicki (1968), "On Schottky groups with applications to kleinian groups", Annals of Mathematics. Second Series 88: 47–61, ISSN 0003-486X, JSTOR 1970555, MR0227403 
• Doyle, Peter (1988), "On the bass note of a Schottky group", Acta Mathematica 160: 249–284, MR945013 
• Fricke, Robert; Klein, Felix (1897) (in German), Vorlesungen über die Theorie der automorphen Functionen. Erster Band; Die gruppentheoretischen Grundlagen., Leipzig: B. G. Teubner, ISBN 978-1-4297-0551-6, JFM 28.0334.01, http://www.archive.org/details/vorlesungenber01fricuoft 
• Fricke, Robert; Klein, Felix (1912) (in German), Vorlesungen über die Theorie der automorphen Functionen. Zweiter Band: Die funktionentheoretischen Ausführungen und die Anwendungen. 1. Lieferung: Engere Theorie der automorphen Funktionen., Leipzig: B. G. Teubner., ISBN 978-1-4297-0552-3, JFM 32.0430.01, http://www.archive.org/details/vorlesungenber02fricuoft 
• Gilman, Jane, A Survey of Schottky Groups, http://www.math.cornell.edu/~vogtmann/MSRI/Gilman%20Notes%20with%20Figures.pdf 
• Jørgensen, T.; Marden, A.; Maskit, Bernard (1979), "The boundary of classical Schottky space", Duke Mathematical Journal 46 (2): 441–446, ISSN 0012-7094, MR534060, http://projecteuclid.org/getRecord?id=euclid.dmj/1077313410 
• Marden, Albert (1974), "The geometry of finitely generated kleinian groups", Annals of Mathematics. Second Series 99: 383–462, ISSN 0003-486X, JSTOR 1971059, MR0349992, Zbl 0282.30014 
• Marden, A. (1977), "Geometrically finite Kleinian groups and their deformation spaces", in Harvey, W. J., Discrete groups and automorphic functions (Proc. Conf., Cambridge, 1975), Boston, MA: Academic Press, pp. 259–293, ISBN 978-0-12-329950-5, MR0494117, http://books.google.com/books?id=gQXvAAAAMAAJ 
• Maskit, Bernard (1967), "A characterization of Schottky groups", Journal d'Analyse Mathématique 19: 227–230, doi:10.1007/BF02788719, ISSN 0021-7670, MR0220929 
• Maskit, Bernard (1988), Kleinian groups, Grundlehren der Mathematischen Wissenschaften, 287, Berlin, New York: Springer-Verlag, ISBN 978-3-540-17746-3, MR959135, http://books.google.com/books?id=qxMzE0-OzrsC 
• David Mumford, Caroline Series, and David Wright, Indra's Pearls: The Vision of Felix Klein, Cambridge University Press, 2002 ISBN 0-521-35253-3
• Schottky, F. (1877), "Ueber die conforme Abbildung mehrfach zusammenhängender ebener Flächen", Journal für die reine und angewandte Mathematik 83: 300–351, ISSN 0075-4102, http://resolver.sub.uni-goettingen.de/purl?GDZPPN002156687 
• Yamamoto, Hiro-o (1991), "An example of a nonclassical Schottky group", Duke Mathematical Journal 63 (1): 193–197, doi:10.1215/S0012-7094-91-06308-8, ISSN 0012-7094, MR1106942 

External links

• Three transformations generating a Schottky group from (Fricke & Klein 1897, p. 442).