  We consider the function $f(g)$ that assigns to an orientable surface $M$ of
genus $g$ the maximal number of free commuting independent involutions on $M$.
We show that the surface of minimal genus $g$ with $f(g)=n$ is a real
moment-angle complex $R_K$, where $K$ is the boundary of an $(n+2)$-gon. The
genus is given by the formula $g = 1 + 2^{n-1}(n-2)$.
