Fricke identities, Frobenius $k$-characters and Markov equation

TL;DR

This paper reveals a deep connection between Frobenius and Fricke's early 20th-century work on group characters and uniformization, linking them to Markov's theory of binary quadratic forms.

Contribution

It uncovers a novel relationship between Frobenius $k$-characters, Fricke's uniformization approach, and Markov's arithmetic of quadratic forms, unifying these areas.

Findings

Link between Frobenius $k$-characters and Fricke's uniformization.

Connection to Markov's work on quadratic forms.

Unified perspective on group theory and number theory.

Abstract

In 1896 Frobenius and Fricke had published two seemingly unrelated papers: Frobenius had started to develop his theory of $k$-characters for finite groups motivated by Dedekind's question about factorisation of the group determinant, while Fricke followed Klein's approach to the uniformization theorem. We show that in fact these two works can be naturally linked and both are related to remarkable Markov's paper of 1880 on arithmetic of binary quadratic forms.