Finite groups acting on torus

Question

Consider the 2-dimensional torus $T^2$ and let $G$ be a finite group acting (faithfully) by diffeomorphisms of $T^2$. Is there a bound for the number of generators of $G$? For example, can it be 5-generated?

Thank you.

Answer 1

There is indeed a bounded number of generators, and it follows from an explicit description of all possible such groups $G$. The idea is that you can make $G$ act by isometries with respect to a Euclidean metric on $T^2$, and then you can lift $G$ via a universal covering map $f : \mathbb R^2 \to T^2$ to obtain a Euclidean crystallographic group $\widetilde G$. These groups are known up to isomorphism: they are the 17 wallpaper groups. Since this is just a finite set of finitely generated groups, one obtains a bound for the number of generators of $\widetilde G$, and since $G$ is just a quotient group of $\widetilde G$ one obtains the same bound for the number of generators of $G$. If you walk through the list of 17, you'll see that each is generated by at most 4 elements.

Notice one thing that this proof does not say: there are not finitely many isomorphism types of the groups $G$, because the kernel of the quotient homomorphism $\widetilde G \mapsto G$ can have arbitrarily large index. What makes this proof work is that the infinite set of possible $G$'s is just a set of quotients of the finite set of groups $\widetilde G$.