Action of free commuting involutions on closed two-dimensional manifolds

Consider a function $f(g)$ associating each oriented surface $M$ of genus $g$ with the maximal number of free commuting involutions on $M$. It is proved that the surface of minimal genus $g$ for which $f(g) = n$ is the moment-angle complex $\mathcal{R}_\mathcal{K}$, where $\mathcal K$ is the boundary of an $(n+2)$-gon. Its genus is given by the formula $g=1+2^{n-1}(n-2)$.