These Thurston maps are NET maps for every choice of translation term. They have degree 4. They are imprimitive, each factoring as a Thurston map with degree 2 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} Since no Thurston multiplier is 1, this modular group Hurwitz class contains only finitely many Thurston equivalence classes. The number of pure modular group Hurwitz classes in this modular group Hurwitz class is 7. ALL THURSTON MULTIPLIERS c/d IN UNREDUCED FORM 0/2, 1/4, 2/1 Every NET map in these pure modular group Hurwitz classes is rational because the modulo 2 correspondence graph has no loops. EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,infinity) SLOPE FUNCTION INFORMATION There are no slope function fixed points because the mod 2 slope correspondence graph has no loops. NONTRIVIAL CYCLES 1/0 -> 0/1 -> 1/0 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 either one of the above cycles or the nonslope. FUNDAMENTAL GROUP WREATH RECURSIONS When the translation term of the affine map is 0: NewSphereMachine( "a=(2,3)", "b=(2,3)", "c=<1,1,1,1>(1,2)(3,4)", "d=(1,2)(3,4)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=(2,3)", "b=(2,3)", "c=<1,1,1,1>(1,2)(3,4)", "d=(1,2)(3,4)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,2)(3,4)", "b=<1,1,1,1>(1,2)(3,4)", "c=(2,3)", "d=(2,3)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(1,2)(3,4)", "b=<1,1,1,1>(1,2)(3,4)", "c=(2,3)", "d=(2,3)", "a*b*c*d");