Fig. (3) The tiling of the Poincar´e plane according to the (tiling) PGL(2,C) group on the desymmetrized domain (smallest tiles) and the tiling according to the modular group (bigger/overlapping tiles) resulting from the composition of (generalized) translation Ta and reflections on circumferences. The tiling according to a modular subgroup of SL(2,C) or SL(2,Z) on the Γ0 domain results of tiling iterations consisting of two tiles here delimited by dashed lines.