Do any two affine rotations with no common fixed point generate an infinite group? Assume we have two affine rotations of the plane around two different fixed points. 
Do they generate an infinite group?

If you have a group $G$ of affine self-transformations of a vector space $V$: then if $G$ is finite then $G$ fixes $\frac1{|G|}\sum_{g\in G}g(0)$. 
By contraposition, if $G$ is generated by a subset $S$ and there is no common fixed point for elements of $S$, then $G$ is infinite.
