When I am reading one paper, I have met the following statement:

It is impossible to define a $Z_{2}\times Z_{2}$ action on a connected closed curve on a compact Riemann surface.

The claim is equivalent to say that the existence of two fixed point free involutions on a circle is not true.

The author just took this statement as an obvious fact and have used it to prove a lemmer about the antiholomorphic involutions on the compact Riemann surface. It seems the claim is pretty simple, but I cannot find an elegant way to prove this is indeed true.

I have tried to use the the following fact to prove it:

the one-dimensional manifold can be considered as a circle under the homeomorphism, and it is diffeomorphic to the one dimensional real projective plane, which inherits the automorphism group of $PGL(2, \mathbb{R})$ from $\mathbb{R}^{2}.$ And in $PGL(2, \mathbb{R}),$ the element corresponding to the antipodal map is unique, so the above statement is true.

Is this a correct proof?

aremany, many fixed-point free involutions on a circle... $\endgroup$