These Thurston maps are NET maps for every choice of translation term. They have degree 9. In fact, they are Euclidean. PURE MODULAR GROUP HURWITZ EQUIVALENCE CLASSES FOR TRANSLATIONS {0,lambda1,lambda2,lambda1+lambda2} This pure modular group Hurwitz class contains infinitely many Thurston equivalence classes. The number of pure modular group Hurwitz classes in this modular group Hurwitz class is 5. ALL THURSTON MULTIPLIERS c/d IN UNREDUCED FORM 1/9, 3/3, 9/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,-0.127451) (-0.127193,infinity ) These Euclidean NET maps are not rational. Every slope function orbit is infinite. FUNDAMENTAL GROUP WREATH RECURSIONS When the translation term of the affine map is 0: NewSphereMachine( "a=(1,8)(2,7)(3,6)(4,5)", "b=(1,8)(2,7)(3,6)(4,5)", "c=(1,9)(2,8)(3,7)(4,6)", "d=(1,9)(2,8)(3,7)(4,6)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=(2,9)(3,8)(4,7)(5,6)", "b=(2,9)(3,8)(4,7)(5,6)", "c=(1,9)(2,8)(3,7)(4,6)", "d=(1,9)(2,8)(3,7)(4,6)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,9)(2,8)(3,7)(4,6)", "b=(1,9)(2,8)(3,7)(4,6)", "c=(1,8)(2,7)(3,6)(4,5)", "d=(1,8)(2,7)(3,6)(4,5)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(1,9)(2,8)(3,7)(4,6)", "b=(1,9)(2,8)(3,7)(4,6)", "c=(2,9)(3,8)(4,7)(5,6)", "d=(2,9)(3,8)(4,7)(5,6)", "a*b*c*d");