Why is the Teichmüller space of a surface homeomorphic to a component of the $\mathrm{PSL} (2, \mathbb R)$ character variety of its fundamental group?

Question

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\PSL{PSL}$ I have a reference request for a proof for the following statement in the title:

The Teichmüller space $T_g$ of the surface $S_g$ of genus $g$ is homeomorphic to a component of the character variety $\Hom (\pi_1 (S_g), \PSL (2, \mathbb R))/\PSL (2, \mathbb R)$.

I understand that Bill Goldman in his PhD thesis ‘Discontinuous groups and the Euler class’ (linked here) proved that a representation is discrete and faithful if and only if its Euler number is maximal, and later in the 1988 paper ‘Topological components of spaces of representations’ (linked here) completely described the components of the character variety as fibres of the Euler number function. However, I want to know how to prove that the component with the maximal Euler number is homeomorphic to Teichmüller space, and Goldman in his papers considers this result classical and does not give an original source for it.

Note that I am not asking what the correspondence between the two spaces is. The bijective correspondence between Teichmüller space and the space of discrete faithful representations up to conjugation can be found in more accessible recent accounts, such as Farb and Margalit's A Primer on Mapping Class Groups. I am asking for a reference for the proof of the assertion that the bijective correspondence is in fact a homeomorphism, where Teichmüller space gets its topology from the Teichmüller metric and the character variety gets its topology naturally as the quotient of a product topology.

I understand that the study of the moduli space of marked hyperbolic structures on surfaces in the character variety flavour was initiated by Fricke, Klein and Nielsen, and can be found in the multivolume work ‘R. Fricke, F. Klein, Vorlesungen über die theorie der automorphen functionen, 1897–1912’. That cannot possibly contain the answer to my question because Teichmüller space and its Teichmüller metric was first defined by O. Teichmüller in late 1930s and early 1940s, much later than work of Fricke, Klein and Nielsen.

Edit log: Changed phrasing from 'original source' to 'reference of a proof'.

Answer 1

I have been told (rather forcefully, regarding this question) that phrases like "original source" or "first example" or "earliest proof" are wrong-headed. The person who scolded me explained (more or less) that all proofs, illustrations, theorems, definitions, etc evolve over time and only slowly converge to their modern forms. So asking for the "original source" is a bit like asking for the date of the "first chicken", or something.

Anyway.

I believe that desired homeomorphism follows from Teichmuller's theorem. The statement you mentioned is (essentially) given as one part of the Corollary at the bottom of page 134 of An introduction to Teichmuller spaces by Imayoshi and Taniguchi. (They use $T_g$ for the "Teichmuller topology" and $F_g$ for the "Fricke topology", which is homeomorphic to the correct component of the character variety.) On page 144 we find:

The original "proof" of Teichmuller's theorem is found in Teichmuller [A-106]. The proof in this chapter follows that in Bers [23].

The reference Bers [23] has the title Quasiconformal mappings and Teichmuller's theorem. I would be remiss if I did not point out Bers' footnote on page 2 of [23]. The author there gently reminds the reader that Teichmuller was an ardent Nazi.