These Thurston maps are NET maps for every choice of translation term. They have degree 8. They are imprimitive, each factoring as a Thurston map with degree 4 followed by a Euclidean NET map with degree 2. The non-Euclidean factor is a NET map if it has four postcritical points. PURE MODULAR GROUP HURWITZ EQUIVALENCE CLASSES FOR TRANSLATIONS {0} {lambda1} {lambda2} {lambda1+lambda2} These pure modular group Hurwitz classes each contain only finitely many Thurston equivalence classes. However, this 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 24. ALL THURSTON MULTIPLIERS c/d IN UNREDUCED FORM 0/4, 1/4, 1/2, 2/4, 2/2 Every NET map in these pure modular group Hurwitz classes is rational because every loop multiplier in the modulo 2 correspondence graph is less than 1. EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,-1.006013) (-0.993807,infinity ) SLOPE FUNCTION INFORMATION There are no slope function fixed points. Number of excluded intervals computed by the fixed point finder: 272 No nontrivial cycles were found. The slope function maps some slope to the nonslope. The slope function orbit of every slope p/q with |p| <= 50 and |q| <= 50 ends in the nonslope. FUNDAMENTAL GROUP WREATH RECURSIONS When the translation term of the affine map is 0: NewSphereMachine( "a=(2,6)(3,7)", "b=(1,8)(2,3)(4,5)(6,7)", "c=(1,7)(2,8)(3,5)(4,6)", "d=(1,2)(3,4)(5,6)(7,8)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=(1,8)(2,3)(4,5)(6,7)", "b=(2,6)(3,7)", "c=<1,1,1,1,1,1,1,1>(1,2)(3,4)(5,6)(7,8)", "d=(1,7)(2,8)(3,5)(4,6)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=<1,1,c^-1,c,1,1,1,1>(1,2)(3,4)(5,6)(7,8)", "b=(1,7)(2,8)(3,5)(4,6)", "c=(1,8)(2,3)(4,5)(6,7)", "d=(2,6)(3,7)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(1,7)(2,8)(3,5)(4,6)", "b=<1,1,1,1,1,1,1,1>(1,2)(3,4)(5,6)(7,8)", "c=(2,6)(3,7)", "d=(1,8)(2,3)(4,5)(6,7)", "a*b*c*d");