Counting homomorphisms from surface groups to finite groups

TL;DR

This paper establishes a formula connecting the count of homomorphisms from surface groups to finite groups with character theory, extending to nonorientable surfaces and applications to symmetric groups.

Contribution

It introduces a unified method to relate homomorphism counts to character sums for various surface types, including nonorientable surfaces.

Findings

Derived a character sum formula for nonorientable surface groups

Extended results to closed and orientable surfaces

Applied the formula to symmetric groups

Abstract

We prove a result that relates the number of homomorphisms from the fundamental group of a compact nonorientable surface to a finite group $G$, where conjugacy classes of the boundary components of the surface must map to prescribed conjugacy classes in $G$, to a sum over values of irreducible characters of $G$ weighted by Frobenius-Schur multipliers. The proof is structured so that the corresponding results for closed and possibly orientable surfaces, as well as some generalizations, are derived using the same methods. We then apply these results to the specific case of the symmetric group.