 The corresponding quotient space Ω("G")/"G" is given by joining up the Jordan curves in pairs, so is a compact Riemann surface of genus "g".This is the boundary of the 3-manifold given by taking the quotient ("H"∪Ω("G"))/"G" of 3-dimensional hyperbolic "H" space plus the regular set Ω("G") by the Schottky group "G", which is a handlebody of genus "g".Conversely any compact Riemann surface of genus "g" can be obtained from some Schottky group of genus "g".