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| In [[mathematics]], a '''free regular set''' is a subset of a [[topological space]] that is acted upon disjointly under a given [[group action]].<ref name=Maskit1987>{{cite book|last=Maskit|first=Bernard|title=Discontinuous Groups in the Plane, Grundlehren der mathematischen Wissenschaften Volume 287|year=1987|publisher=Springer Berlin Heidelberg|isbn=978-3-642-64878-6|pages=15–16}}</ref>
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| To be more precise, let ''X'' be a [[topological space]]. Let ''G'' be a group of [[homeomorphism]]s from ''X'' to ''X''. Then we say that the action of the group ''G'' at a point <math>x\in X</math> is '''freely discontinuous''' if there exists a [[Neighbourhood (mathematics)|neighborhood]] ''U'' of ''x'' such that <math>g(U)\cap U=\varnothing</math> for all <math>g\in G</math>, excluding the identity. Such a ''U'' is sometimes called a ''nice neighborhood'' of ''x''.
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| The set of points at which G is freely discontinuous is called the '''free regular set''' and is sometimes denoted by <math>\Omega=\Omega(G)</math>. Note that <math>\Omega</math> is an [[open set]].
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| If ''Y'' is a subset of ''X'', then ''Y''/''G'' is the space of equivalence classes, and it inherits the canonical topology from ''Y''; that is, the projection from ''Y'' to ''Y''/''G'' is continuous and open.
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| Note that <math>\Omega /G</math> is a [[Hausdorff space]].
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| ==Examples==
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| The open set
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| :<math>\Omega(\Gamma)=\{\tau\in H: |\tau|>1 , |\tau +\overline\tau| <1\}</math>
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| is the free regular set of the [[modular group]] <math>\Gamma</math> on the [[upper half-plane]] ''H''. This set is called the [[fundamental domain]] on which [[modular form]]s are studied.
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| ==See also==
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| * [[Covering map]]
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| * [[Klein geometry]]
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| * [[Homogenous space]]
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| * [[Clifford–Klein form]]
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| * [[G-torsor]]
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| ==References==
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| {{reflist}}
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| {{DEFAULTSORT:Free Regular Set}}
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| [[Category:Topological groups]]
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| [[Category:Group actions]]
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Latest revision as of 03:15, 2 December 2014
Hello. Let me introduce the author. Her title is Refugia Shryock. California is our beginning location. It's not a typical factor but what she likes doing is base leaping and now she is trying to earn cash with it. Hiring is his profession.
my site: std testing at home