Why a discrete group of diffeomorphisms $\mathbb S^2\rightarrow\mathbb S^2$ can't have exactly one fixed point? Context: while reading Thurston's notes on the geometry of 3-manifolds, I have found the assertion that the orbifold obtained from a sphere by adding a single conic point is a bad orbifold. However, his argument relies on the nonexistence of a discrete group of diffeomorphisms of the sphere fixing exactly one point.
How can we prove this? I thought that there should be an easy argument involving properties of degrees of maps between spheres that should clarify the situation, but I have not found it.
Thanks in advance for your answers.
 A: We'll follow Moishe Kohan's indication.
Suppose that the teardrop orbifold is indeed a good orbifold. That means, after realizing that the only orbifold covers of a cone is a cone of order dividing that of the original one, that the universal covering orbifold of the tesrdrop is a sphere, and that it can be realized as a quotient of the sphere by a discrete group of diffeomorphisms. Now the isotropy group of the cone point $x$ is finite, and the restriction of all the diffeomorphisms that fix $x$ to $\mathbb S^2\setminus\{x\}$ determines, after composing them with the stereographic projection, a finite group of diffeomorphisms of the plane. After choosing an element of prime order $f$, a classical theorem of Smith then assures that the group generated by $f$ must fix a point of $\mathbb R^2$. But then, pulling back the group $\langle f\rangle$ by precomposing with the inverse of the stereographic projection gives rise to a subgroup of the isotropy group of $x$ that fixes another point of the sphere, a contradiction.
