Does there exist such a representation? In the title, $\Sigma_{2}$ means the closed orientable surface of genus 2.
I once heard of this or something like it, but not quite sure. Thanks to everyone!
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1$\begingroup$ No doubt there's an explicit construction, but perhaps it's worth noting that $PSL(2,\mathbb{Z}[i])$ is virtually the fundamental group of a non-compact hyperbolic 3-manifold of finite volume, and Cooper--Long--Reid proved that all such 3-manifold groups contain surface subgroups. I'm not sure whether it's clear that their construction should give genus two. I think that one needs to know that some element of the cusp subgroup bounds a punctured torus. $\endgroup$– HJRWNov 12, 2015 at 21:56
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$\begingroup$ You can actually find a lot of locally totally geodesic imbeddings (which are automatically $\pi_1$-injective) of closed hyperbolic 2-orbifolds into the orbifold associated to $\mathrm{PSL}_2(\mathbb{Z}[i])$. Any division quaternion over $\mathbb{Q}$ that splits over $\mathbb{Q}(i)$ gives a family of such. For more detail you can consult Chapter 9.5 in MacLachlan--Reid, The arithmetic of hyperbolic 3--manifolds. $\endgroup$– Jean RaimbaultNov 14, 2015 at 15:34
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$\begingroup$ You can then explicitely compute the volume of the imbedded orbifolds from the arithmetic data for the quaternion algebra (see Chapter 11 in loc. cit.). It seems likely you can find one small enough so that it is covered by a genus 2 surface, which would answer your question in the affirmative. $\endgroup$– Jean RaimbaultNov 14, 2015 at 15:37
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