I'm learning Riemann surfaces with Donaldson' Riemann surfaces.
In Donaldson, he proved that if $X$ is Riemannian surfaces and $\Gamma$ is subgroup of group of holomorphic automorphism of $X$(that is, $PSL(2,\mathbb{R})$) acts freely on $X$.
I want to find concrete non trivial example of $\Gamma$ but didn't.
For example let $\Gamma = \bigg\langle\begin{pmatrix} 2 & 1 \\ 1 &1\end{pmatrix}\bigg\rangle$, how I know whether this acts freely or not?
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In previous example now I see this acts freely since 2+1>2. Just I didn't catch it. But It's too hard to visualize, so is there any 'nice' example?
A very straightforward example is the group generated by $T=\pmatrix{1&1\\0&1}$. This is the map $z\mapsto z+1$, and so has no fixed points. A disc $D$ with Euclidean diameter $<1$ is mapped by the powers of $T$ into a sequence $(T^n(D))_{n=-\infty}^\infty$ of disjoint discs, which makes it clear that the action is free.