Convergence group

<p>In mathematics, a convergence group or a discrete convergence group is a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> 
  
    
      
        Γ
      
    
    {\displaystyle \Gamma }
  
 <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">acting</a> by <a href="/facts/Homeomorphism/jMfWg3Mn">homeomorphisms</a> on a <a href="/facts/Compact_space/d0cgXJH7">compact</a> <a href="/facts/Metrizable_space/BCLdo5Jf">metrizable space</a> 
  
    
      
        M
      
    
    {\displaystyle M}
  
 in a way that generalizes the properties of the action of <a href="/facts/Kleinian_group/FBMCx9Ry">Kleinian group</a> by <a href="/facts/M%25C3%25B6bius_transformation/YEkTTLfK">Möbius transformations</a> on the ideal boundary 
  
    
      
        
          
            S
          
          
            2
          
        
      
    
    {\displaystyle \mathbb {S} ^{2}}
  
 of the <a href="/facts/Hyperbolic_space/kI7tnwdk">hyperbolic 3-space 
  
    
      
        
          
            H
          
          
            3
          
        
      
    
    {\displaystyle \mathbb {H} ^{3}}
  
</a>.
The notion of a convergence group was introduced by <a href="/facts/Frederick_Gehring/o3lpPH01">Gehring</a> and <a href="/facts/Gaven_Martin/4NjEK2cJ">Martin</a> (1987)  and has since found wide applications in <a href="/facts/Geometric_topology/NpsqSK2N">geometric topology</a>, <a href="/facts/Quasiconformal_mapping/gICCqYEb">quasiconformal analysis</a>, and <a href="/facts/Geometric_group_theory/AJhxpFUe">geometric group theory</a>.
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Convergence group open-in-new

Convergence group