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| In mathematics, a '''Schottky group''' is a special sort of [[Kleinian group]], first studied by {{harvs|txt|first=Friedrich |last = Schottky|authorlink=Friedrich Schottky|year=1877}}.
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| ==Definition==
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| Fix some point ''p'' on the [[Riemann sphere]]. Each [[Jordan curve]] not passing through ''p''
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| 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".
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| Suppose there are 2''g'' disjoint [[Jordan curve]]s ''A''<sub>1</sub>, ''B''<sub>1</sub>,..., ''A''<sub>''g''</sub>, ''B''<sub>''g''</sub> in the [[Riemann sphere]] with disjoint interiors.
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| If there are [[Moebius transformation]]s ''T''<sub>''i''</sub> taking the outside of ''A''<sub>''i''</sub> onto the inside of ''B''<sub>''i''</sub>, then the group generated by these transformations is a [[Kleinian group]]. A '''Schottky group''' is any Kleinian group that can be constructed like this.
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| ==Properties==
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| Schottky groups are [[generating set of a group|finitely generated]] [[free group]]s such that all non-trivial elements are [[Möbius transformation#Classification|loxodromic]]. Conversely {{harvtxt|Maskit|1967}} showed that any finitely generated free Kleinian group such that all non-trivial elements are loxodromic is a Schottky group.
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| 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''.
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| ==Classical and non-classical Schottky groups==
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| A Schottky group is called '''classical''' if all the disjoint Jordan curves corresponding to some set of generators can be chosen to be circles. {{harvs|txt|last=Marden|year1=1974|year2=1977}} gave an indirect and non-constructive proof of the existence of non-classical Schottky groups, and {{harvtxt|Yamamoto|1991}} gave an explicit example of one. It has been shown by {{harvtxt|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. Conversely, {{harvtxt|Hou|2010}} has proved that there exists a universal lower bound on the Hausdorff dimension of limit sets of all non-classical Schottky groups.
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| ==Limit sets of Schottky groups==
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| The [[limit set]] of a Schottky group, the complement of Ω(''G''), always has [[Lebesgue measure]] zero, but can have positive ''d''-dimensional [[Hausdorff measure]] for ''d'' < 2. It is perfect and nowhere dense with positive logarithmic capacity.
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| The statement on Lebesgue measures follows for classical Schottky groups from the existence of the [[Poincaré series (modular form)|Poincaré series]]
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| :<math>\displaystyle{P(z)=\sum (c_iz+d_i)^{-4}.}</math>
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| [[Henri Poincaré|Poincaré]] showed that the series | ''c''<sub>''i''</sub> |<sup>–4</sup> is summable over the non-identity elements of the group. In fact taking a closed disk in the interior of the fundamental domain, its images under different group elements are disjoint and contained in a fixed disk about 0. So the sums of the areas is finite. By the changes of variables formula, the area is greater than a constant times | ''c''<sub>''i''</sub> |<sup>–4</sup>.<ref>{{harvnb|Lehner|1964|p=159}}</ref>
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| A similar argument implies that the limit set has Lebesgue measure zero.<ref>{{harvnb|Akaza|1963}}</ref> For it is contained in the complement of union of the images of the fundamental region by group elements with word length bounded by ''n''. This is a finite union of circles so has finite area. That area is bounded above by a constant times the contribution to the Poincaré sum of elements of word length ''n'', so decreases to 0.
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| ==Schottky space==
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| 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 PSL<sub>2</sub>('''C''') that generate a Schottky group, up to equivalence under Moebius transformations {{harv|Bers|1975}}. It is a complex manifold of complex dimension 3''g''−3. It contains classical Schottky space as the subset corresponding to classical Schottky groups.
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| 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.
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| ==See also==
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| *[[Beltrami_equation#Uniformization_of_multiply_connected_planar_domains|Beltrami equation]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{citation|last=Akaza|first= Tohru|title=Poincaré theta series and singular sets of Schottky groups|journal=Nagoya Math. J. |volume=24|year= 1964|pages= 43–65}}
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| *{{Citation | last1=Bers | first1=Lipman | title=Automorphic forms for Schottky groups | doi=10.1016/0001-8708(75)90117-6 | mr=0377044 | year=1975 | journal=Advances in Mathematics | issn=0001-8708 | volume=16 | pages=332–361}}
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| *{{Citation | last1=Chuckrow | first1=Vicki | title=On Schottky groups with applications to kleinian groups | jstor=1970555 | mr=0227403 | year=1968 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=88 | pages=47–61}}
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| *{{Citation | last1=Doyle | first1=Peter | title=On the bass note of a Schottky group| mr=945013 | year=1988 | journal=[[Acta Mathematica]] | volume=160 | pages=249–284}}
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| *{{Citation | last1=Fricke | first1=Robert | last2=Klein | first2=Felix | title=Vorlesungen über die Theorie der automorphen Functionen. Erster Band; Die gruppentheoretischen Grundlagen. | url=http://www.archive.org/details/vorlesungenber01fricuoft | publisher=Leipzig: B. G. Teubner | language=German | isbn=978-1-4297-0551-6 | jfm=28.0334.01 | year=1897}}
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| *{{Citation | last1=Fricke | first1=Robert | last2=Klein | first2=Felix | title=Vorlesungen über die Theorie der automorphen Functionen. Zweiter Band: Die funktionentheoretischen Ausführungen und die Anwendungen. 1. Lieferung: Engere Theorie der automorphen Funktionen. | url=http://www.archive.org/details/vorlesungenber02fricuoft | publisher=Leipzig: B. G. Teubner. | language=German | isbn=978-1-4297-0552-3 | jfm=32.0430.01 | year=1912}}
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| *{{citation| last=Gilman|first=Jane|url=http://www.math.cornell.edu/~vogtmann/MSRI/Gilman%20Notes%20with%20Figures.pdf |title=A Survey of Schottky Groups}}
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| *{{Citation | last1=Hou | first1=Yong | title=Kleinian groups of small Hausdorff dimension are classical Schottky groups I| year=2010 | journal=[[Geometry & Topology]] | volume=14 | pages=473–519}}
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| *{{Citation | last1=Hou | first1=Yong | title=All finitely generated Kleinian groups of small Hausdorff dimension are classical Schottky groups|url=http://arxiv.org/abs/1307.2677}}
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| *{{Citation | last1=Jørgensen | first1=T. | last2=Marden | first2=A. | last3=Maskit | first3=Bernard | title=The boundary of classical Schottky space | url=http://projecteuclid.org/getRecord?id=euclid.dmj/1077313410 | mr=534060 | year=1979 | journal=[[Duke Mathematical Journal]] | issn=0012-7094 | volume=46 | issue=2 | pages=441–446}}
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| *{{citation|series=Mathematical Surveys and Monographs|year=1964|volume= 8|id=ISBN 0-8218-1508-3|title=Discontinuous Groups and Automorphic Functions|first=Joseph|last= Lehner|publisher=American Mathematical Society}}
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| *{{Citation | last1=Marden | first1=Albert | title=The geometry of finitely generated kleinian groups | jstor=1971059 | mr=0349992 | zbl = 0282.30014 | year=1974 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=99 | pages=383–462}}
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| *{{Citation | last1=Marden | first1=A. | editor1-last=Harvey | editor1-first=W. J. | title=Discrete groups and automorphic functions (Proc. Conf., Cambridge, 1975) | url=http://books.google.com/books?id=gQXvAAAAMAAJ | publisher=[[Academic Press]] | location=Boston, MA | isbn=978-0-12-329950-5 | mr=0494117 | year=1977 | chapter=Geometrically finite Kleinian groups and their deformation spaces | pages=259–293}}
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| *{{Citation | last1=Maskit | first1=Bernard | title=A characterization of Schottky groups | doi=10.1007/BF02788719 | mr=0220929 | year=1967 | journal=Journal d'Analyse Mathématique | issn=0021-7670 | volume=19 | pages=227–230}}
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| *{{Citation | last1=Maskit | first1=Bernard | title=Kleinian groups | url=http://books.google.com/books?id=qxMzE0-OzrsC | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften | isbn=978-3-540-17746-3 | mr=959135 | year=1988 | volume=287}}
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| *[[David Mumford]], Caroline Series, and David Wright, ''[[Indra's Pearls (book)|Indra's Pearls: The Vision of Felix Klein]]'', [[Cambridge University Press]], 2002 ISBN 0-521-35253-3
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| *{{Citation | last1=Schottky | first1=F. | title=Ueber die conforme Abbildung mehrfach zusammenhängender ebener Flächen | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002156687 | year=1877 | journal=[[Journal für die reine und angewandte Mathematik]] | issn=0075-4102 | volume=83 | pages=300–351}}
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| *{{Citation | last1=Yamamoto | first1=Hiro-o | title=An example of a nonclassical Schottky group | doi=10.1215/S0012-7094-91-06308-8 | mr=1106942 | year=1991 | journal=[[Duke Mathematical Journal]] | issn=0012-7094 | volume=63 | issue=1 | pages=193–197}}
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| ==External links==
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| *[http://www.archive.org/stream/vorlesungenber01fricuoft#page/442/mode/2up Three transformations generating a Schottky group] from {{harv|Fricke|Klein|1897|p= 442}}.
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| [[Category:Kleinian groups]]
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| [[Category:Group theory]]
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| [[Category:Discrete groups]]
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| [[Category:Lie groups]]
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The author is called Wilber Pegues. Credit authorising is how she makes a residing. I've usually loved living in Alaska. To climb is something I truly appreciate doing.
Feel free to surf to my blog post - real psychic, appin.co.kr,