Elementary Groups and Jörgensen’s Inequality

Clearly, any subgroup in which all elements share some fixed point is elementary. We’ve noted how an element of PSL(2,R) takes elements from H to H. However, it follows directly from the closure of the real numbers that PSL(2,R) takes elements from R to R. If we look at the limiting behavior when dealing with ∞, we can say that PSL(2,R) takes elements from R ⋃ {∞} to R ⋃ {∞}. In other words, orbits stay in R ⋃ {∞} or in H, though in either case the group acts on the closure of H.