A generating function of the number of homomorphisms from a surface group into a finite group

Abstract

A generating function of the number of homomorphisms from the fundamental group of a compact oriented or non-orientable surface without boundary into a finite group is obtained in terms of an integral over a real group algebra. We calculate the number of homomorphisms using the decomposition of the group algebra into irreducible factors. This gives a new proof of the classical formulas of Frobenius, Schur, and Mednykh.

12 pages, 1 figure. Prepared in AMS-LaTeX

References in corpus (6)

• Chern-Simons Theory with Finite Gauge Group
• Gromov-Witten theory, Hurwitz numbers, and Matrix models, I
• Ribbon Graphs, Quadratic Differentials on Riemann Surfaces, and Algebraic Curves Defined over $\bar{Q}$
• The equivariant Gromov-Witten theory of P^1
• Random Matrices and Random Permutations
• On the distribution of the length of the second row of a Young diagram under Plancherel measure

Cited by in corpus (5)

• On 2D gauge theories in Jackiw-Teitelboim gravity
• Volume of representation varieties
• Geometry of Character Varieties of Surface Groups
• Mednykh's Formula via Lattice Topological Quantum Field Theories
• Counting homomorphisms from surface groups to finite groups