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Suppose that $S$ is a closed connected surface with negative Euler characteristic. Suppose that $H$ is the universal cover of $S$. Equipping $S$ with a hyperbolic metric we find that $H$ is isometric to the hyperbolic plane. Exercise: Every non-trivial deck transformation of $H$ over $S$ is hyperbolic: that is, has exactly two fixed points at infinity. ...
I recently returned to this question, and have found a formal solution to Closing the Steinberg symbol in genus two using Mark C. Bell's wonderful curver program. In low genus it was expected that an interesting finite order subgroup could yield formal solutions. This is the case in $PGL(\mathbb{Z}^2)$. For the mapping class group of genus two, there is an ...