Via the uniformisation theorem this gives rise to a representation of its fundamental group in the isometry group of the hyperbolic plane , which is isomorphic to PSL ( R ) .
This realises the fundamental group as a Fuchsian group .
Via the uniformisation theorem this gives rise to a representation of its fundamental group in the isometry group of the hyperbolic plane , which is isomorphic to PSL ( R ) and this realises the fundamental group as a Fuchsian group .