# Why is the Fuchsian group of the first kind not cyclic?

a Fuchsian group is a discrete subgroup of $$\text{PSL}_2(\mathbb R)$$, which we can view as a group of transformations of the upper half-plane $$\mathbb H$$ that acts discontinuously.

A Fuchsian group $$\Gamma$$ is said to be of the first kind if every point in $$\mathbb R\cup\{\infty\}$$ (the boundary of $$\mathbb H$$) is a limit point of the orbit $$\Gamma z$$ for some $$z\in\mathbb H$$. Here, the notion of "limit point" is with respect to the topology on the Riemann sphere $$\mathbb C_\infty$$.

Why is a Fuchsian group of the first kind not cyclic? I understand that it can't be finite. But it's hard to rule out the infinite cyclic case.

Source: Iwaniec's Spectral methods in automorphic forms

If $$g$$ is elliptic, then the group generated by $$g$$ is finite.
If $$g$$ is parabolic then there is only one limit point for $$g^n z$$, independent of $$z$$. For example if $$g(z) = z+1$$, then this limit point is $$\infty$$ regardless of what $$z$$ we start at. So then the group generated by $$g$$ is not Fuchsian of the first kind.
If $$g$$ is hyperbolic, then there are only 2 limit points for $$g^n z$$, independent of $$z$$. For example, if $$g(z) = 2z$$ then these limit points are 0 and $$\infty$$. So again the group generated by $$g$$ is not Fuchsian of the first kind.