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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

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An element of a Fuchsian group is either elliptic, parabolic, or hyperbolic.

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.

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