# Free commuting involutions on closed two-dimensional surfaces

## TL;DR

This paper investigates the maximum number of free commuting involutions on orientable surfaces, identifying minimal genus surfaces with a given number of involutions as specific real moment-angle complexes.

## Contribution

It establishes a precise relationship between the number of involutions and the minimal genus of surfaces, characterizing these surfaces as real moment-angle complexes derived from polygons.

## Key findings

- Minimal genus surfaces with n involutions are real moment-angle complexes.
- The genus of these surfaces is given by g = 1 + 2^{n-1}(n-2).
- Identifies the boundary of an (n+2)-gon as key in the surface structure.

## Abstract

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 $g = 1 + 2^{n-1}(n-2)$.