# example of fuchsian groups acting on 2-sphere by G. Martin

Currently I am reading a paper "Infinite group actions on spheres" by Gaven Martin. I am a first year graduate students and I got lots of questions, so one of them is about the following example: (sorry in advance, if that question would be trivial)

In Example 4.2. of that paper, we let $G$ be a nonelementary Fuchsian group acting on $\mathbb{S}^2$. Then $G$ leaves the disk $D^2$ invariant, and we identify the disk to a point $x_0$ and extend the action of $G$ over this point by agreeing that every element of $G$ fixes $x_0$. This produces a group homeomorphism $H$ of $\mathbb{S}^2 / D \cong \mathbb{S}^2$ acting properly discontinuously in the complement of the point $x_0$. Thus $H$ is a convergence group and is not conjugate to any Mobius group since the stabilizer of a point in a Mobius group is virtually abelian and the stabilizer of $x_0$ is $H \cong G$.

$\textbf{Now my question:}$ As I understand, the $G$-invariant disk $D^2$ should be considered as a subset of $\mathbb{S}^2$, for example the upper hemisphere, and by definition of Fuchsian group $G$ acts properly discontinuously on $D^2$. But after the identification it to $x_0$, the complement of $x_0$ is $\mathbb{S}^2 - D^2$, and the action of $G$ on $\mathbb{S}^2 - D^2$ is not propery discontinuous because $G$ is nonelementary and $\mathbb{S}^2 - D^2$ contains some limit points. Since $H$ is produced from $G$ by $$H|_{\mathbb{S}^2 - D^2} = G|_{\mathbb{S}^2 - D^2} \quad \textrm{and} \quad Hx_0 = x_0,$$ then the action of $H$ on the complement of $x_0$ is not properly discontinuous too.

This is completely contrast to the initial example by G. Martin. Can someone show me where the mistake in my thoughts is?

• The limits points of a Fuchsian group are all on the boundary of the disk. Therefore, if we take $D^2$ to be the closed disk, then, contrary to what you say, there are no limit points in $\mathbb{S}^2 - D^2$. If $H$ is elementary, then it is virtually abelian, so there is no contradiction at the end of the proof. – Dave Witte Morris Jul 17 '16 at 22:26
• @DaveWitteMorris thank you very much! just one more question: if we let $G$ be a nonelementary Fuchsian group acting on closed disk in complex plane. We identify the boundary circle to point $x_0$, and we get a $2$-sphere. Then the produced group $H$ fixes $x_0$ and acts on its complement properly discontinuously. and the result would be the same, am I right?? Thank you in advance! – Jane Jul 18 '16 at 0:48
• Yes, that's another way of doing the same thing as Martin (except that you replaced the complement of $D^2$ with the interior of $D^2$). – Dave Witte Morris Jul 18 '16 at 3:49