= Injection of a finite hyperbolic isometry group into a mapping class group
If a finite group $G$ acts faithfully by orientation-preserving isometries on a closed hyperbolic surface $S$ of genus at least two, then $G\hookrightarrow\operatorname{Mod}(S)$. An isometry isotopic to the identity induces an inner automorphism of the surface group; its suitable lift commutes with every deck transformation and fixes their limit set, forcing the lift and the original isometry to be the identity.
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