Discontinuous group

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

Definition

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