Linear fractional group as Galois group

TL;DR

This paper classifies all orientation-preserving actions of certain projective special linear groups on orientable surfaces by computing their signatures, linking group theory, topology, and algebraic geometry.

Contribution

It provides a complete classification of signatures for $PSL_2({7})$ and $PSL_2({11})$, connecting group actions with Galois theory and geometric structures.

Findings

Classified all signatures for $PSL_2({7})$ and $PSL_2({11})$

Linked group actions to the inverse Galois problem

Bridged topology, geometry, and algebra in the classification

Abstract

We compute all signatures of $PSL_2(\mathbb{F}_7)$, and $PSL_2(\mathbb{F}_{11})$ which classify all orientation preserving actions of the groups $PSL_2(\mathbb{F}_7)$, and $PSL_2(\mathbb{F}_{11})$ on compact, connected, orientable surfaces with orbifold genus $\geq 0$. This classification is well-grounded in the other branches of Mathematics like topology, smooth, and conformal geometry, algebraic categories, and it is also directly related to the inverse Galois problem.