A Theorem by Gallo, Goldman and Porter states the following:

Let $S_g$ be a closed orientable surface of genus $g$ with fundamental group $\Gamma_g$, and fix a non-elementary representation $\rho\colon \Gamma_g \to PSL(2,\mathbb{R})$. Then, there exists a pants decomposition of $S_g$ whose curves are all taken by $\rho$ into hyperbolic elements.

I am looking for a reference to a proof of this statement. I know that the above result is a corollary to the results proved in Sections 3-4-5 of

Gallo, Kapovich, Marden, "The monodromy groups of Schwarzian equations on closed Riemann surfaces", Ann. of Math. (2) 151 (2000), no. 2, 625–704.

However, that paper deals with the more general case of representations into $PSL(2,\mathbb{C})$, so I expect that the proof in the real case could be shorter.

As stated by Tan in "Branched CP1-structures on surfaces with prescribed real holonomy", Math. Ann. 300 (1994), no. 4, 649–667,

there is a proof of the theorem above result in

D.M. Gallo, W.M. Goldman, R.M. Porter, "Projective structures with monodromy in PSL(2,R)", preprint.

However, I am not able to find this preprint. When looking for that preprint, MathSciNet redirects to the paper

Goldman, "Projective structures with Fuchsian holonomy", J. Differential Geom. 25 (1987), no. 3, 297–326,

but I am not sure that the statement I am interested in is proved there.

  • $\begingroup$ Roberto: I have their preprint, the proof they had was about the same length as ours. There is also an argument which mostly reduces complex case to the real one. The key is that a point in $SL(2,C)$-character variety is determined by traces of simple loops. Incidentally, there are nonelementary $SL(2,R)$-representations of 4-times punctured sphere groups, where every simple loop has elliptic image. $\endgroup$ – Misha Jul 17 '13 at 15:44
  • $\begingroup$ Dear Misha, thank you very much for the information. So it is probably not worth going for their proof. I suspected that the statement could be false for punctured spheres (even if I did not try to construct any example). $\endgroup$ – Roberto Frigerio Jul 17 '13 at 22:01

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