Weak localization

2014-01-27T13:50:04Z

129.10.128.113:

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}}.

==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 2''g'' disjoint [[Jordan curve]]s ''A''1, ''B''1,..., ''A''''g'', ''B''''g'' in the [[Riemann sphere]] with disjoint interiors.
If there are [[Moebius transformation]]s ''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 [[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.

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. {{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.

==Limit sets of Schottky groups==
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.

The statement on Lebesgue measures follows for classical Schottky groups from the existence of the [[Poincaré series (modular form)|Poincaré series]]

:<math>\displaystyle{P(z)=\sum (c_iz+d_i)^{-4}.}</math>

[[Henri Poincaré|Poincaré]] showed that the series | ''c''''i'' |–4 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''''i'' |–4.<ref>{{harvnb|Lehner|1964|p=159}}</ref>

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.

==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 {{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.

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.

==See also==
*[[Beltrami_equation#Uniformization_of_multiply_connected_planar_domains|Beltrami equation]]
==Notes==
{{reflist}}

==References==
*{{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}}
*{{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}}
*{{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}}
*{{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}}
*{{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}}
*{{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}}
*{{citation| last=Gilman|first=Jane|url=http://www.math.cornell.edu/~vogtmann/MSRI/Gilman%20Notes%20with%20Figures.pdf |title=A Survey of Schottky Groups}}
*{{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}}
*{{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}}
*{{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}}
*{{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}}
*{{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}}
*{{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}}
*{{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}}
*{{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}}
*[[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
*{{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}}
*{{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}}

==External links==
*[http://www.archive.org/stream/vorlesungenber01fricuoft#page/442/mode/2up Three transformations generating a Schottky group] from {{harv|Fricke|Klein|1897|p= 442}}.

[[Category:Kleinian groups]]
[[Category:Group theory]]
[[Category:Discrete groups]]
[[Category:Lie groups]]

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Weak localization