Discontinuous group

Short description: Mathematical concept

A discontinuous group is a mathematical concept relating to mappings in topological space.

Definition

Let [math]\displaystyle{ T }[/math] be a topological space of points [math]\displaystyle{ \tau }[/math], and let [math]\displaystyle{ \tau\to f(\tau,x) }[/math], [math]\displaystyle{ x\in G }[/math], be an open continuous representation of the topological group [math]\displaystyle{ G }[/math] as a transitive group of homeomorphic mappings of [math]\displaystyle{ T }[/math] on itself. The representation [math]\displaystyle{ \tau\to f(\tau,a) }[/math] [math]\displaystyle{ a\in H }[/math] of the discrete subgroup [math]\displaystyle{ H\sub G }[/math] in [math]\displaystyle{ T }[/math] is called discontinuous, if no sequence [math]\displaystyle{ f(\tau,a_n) }[/math] ([math]\displaystyle{ n=1,2,\ldots }[/math]) converges to a point in [math]\displaystyle{ T }[/math], as [math]\displaystyle{ a_n }[/math] runs over distinct elements of [math]\displaystyle{ H }[/math].