I am working with the book "Fuchsian groups", by Svetlana Katok and am trying to solve one of the given exercises in chapter 2.
The goal is to prove that every hyperbolic and parabolic cyclic subgroup of $PSL(2,\mathbb{R})$ is Fuchsian.
For this, I would like to show that the following are discrete:
- When one conjugates any hyperbolic element to $z\mapsto kz$ for some $k>0$.
- When one conjugates any parabolic element to $z\mapsto z+1$ or $z\mapsto z-1$
However, I don't understand how to show that these are discrete, any help would be greatly appreciated.
This should help you to get started. The other cases follow the same idea.
Let $\Gamma=\langle T\rangle$ be a hyperbolic cyclic subgroup of $PSL(2,\mathbb{R})$. T being hyperbolic means then there exists some $S\in PSL(2,\mathbb{R})$ such that $$T=S\begin{pmatrix} \lambda & 0 \\ 0 & \frac{1}{\lambda} \end{pmatrix} \quad S^{-1}$$ So then you also have: $$T^{n}=S\begin{pmatrix} \lambda^{n} & 0 \\ 0 & \frac{1}{\lambda^{n}} \end{pmatrix} \quad S^{-1}$$ Now for the discreteness, you need to try and find some sequence, say $\{T_{k}\}$ which converges to the identity. You can do so by using the above.