TITLE: When do the lengths of simple closed curves determine a hyperbolic surface?
QUESTION [9 upvotes]: Consider hyperbolic metrics on $\Sigma_g$ a closed orientable surface of genus $g$. Let $[\gamma_1] , \cdots, [\gamma_n]$ be a finite collection of isotopy classes of simple closed curves on $\Sigma_g$.
At what conditions on $[\gamma_1] , \cdots, [\gamma_n]$ is any (isotopy class of) metric completely determined by the lengths of the geodesic representatives of respective classes $[\gamma_1] , \cdots, [\gamma_n]$ ?
I am not necessarily looking for a complete answer and partial answers/comments are very welcome.

REPLY [5 votes]: This paper is probably relevant for your question:
MR0528966
Wolpert, Scott
The length spectra as moduli for compact Riemann surfaces.
Ann. of Math. (2) 109 (1979), no. 2, 323–351.
And here is a more recent paper on the subject:
MR3770180
Parlier, Hugo
Interrogating surface length spectra and quantifying isospectrality. (English summary)
Math. Ann. 370 (2018), no. 3-4, 1759–1787.
These papers suggest that finitely many geodesic lengths never determine the surface.