Free regular set: Difference between revisions
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#REDIRECT [[Group action#Topological properties]] |
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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 (mathematics)|group action]].<ref name=Maskit1987>{{cite book|last=Maskit|first=Bernard|authorlink=Bernard Maskit|title=Discontinuous Groups in the Plane |series=Grundlehren der mathematischen Wissenschaften |volume=287|year=1987|publisher=Springer |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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* [[Homogeneous 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 (mathematics)]] |
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Latest revision as of 08:08, 17 August 2022
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