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The following question may be trivial or inappropriate; I am not sure though.

It is known that a cocompact oriented Fuchsian group $\Gamma$ admits a presentation: for given $m,g,d_i \geq 0$ $$ \Gamma := \langle x_1, \ldots, x_m , y_1, \ldots, y_g, z_1, \ldots, z_g | x_1^{d_1}, \ldots ,x_m^{d_m}, x_1 \ldots x_m [y_1,z_1] \ldots [y_g,z_g] \rangle $$ In the case where $m =0$ and $g > 1$ one obtains surface group of genus $g$. The question is whether there are homomorphisms with interesting properties between Fuchsian groups and surface groups, with the same genus $g$?

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  • $\begingroup$ There's the retraction obtained by mapping all $x_i$ to 1. You can also compose (on the surface group side) by elements in the mapping class group/$\mathrm{Out}(\pi_1(\Sigma_g))$. Beyond that I don't see anything "natural" without changing genus. $\endgroup$ – Jean Raimbault Feb 19 '17 at 8:13
  • $\begingroup$ Thank you. Are there any maps if the genus differs? $\endgroup$ – user49908 Feb 19 '17 at 8:26
  • $\begingroup$ You get inclusion of a larger genus surface group into any cocompact Fuchsian group by "Selbergs lemma" (which probably has an easy topological proof in this case). $\endgroup$ – Jean Raimbault Feb 19 '17 at 11:10
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    $\begingroup$ If the image has infinite index then it is virtually free. In the surface case this means that all of these maps factor through a free group of genus g (Makanin-Razborov diagram for surface groups). I would assume that there is a similar statement for understanding all homomorphisms from a fixed Fuchsian group into all virtually free groups. $\endgroup$ – Richard Weidmann Feb 20 '17 at 18:40
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    $\begingroup$ ... that occur as subgroups of some other fixed Fuchsian group. $\endgroup$ – Richard Weidmann Feb 20 '17 at 19:04

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