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} These pure modular group Hurwitz classes each contain infinitely many Thurston equivalence classes. The number of pure modular group Hurwitz classes in this modular group Hurwitz class is 8. ALL THURSTON MULTIPLIERS c/d IN UNREDUCED FORM 0/1, 1/4, 2/2, 2/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,-0.010000) ( 0.010000, infinity) The half-space computation does not determine rationality. EXCLUDED INTERVALS FOR JUST THE SUPPLEMENTAL HALF-SPACE COMPUTATION INTERVAL COMPUTED FOR HST OR EXTENDED HST (-0.500000,0.500000) 0/1 EXTENDED HST The supplemental half-space computation shows that these NET maps are rational. SLOPE FUNCTION CYCLES FOUND NUMBER OF FIXED POINTS FOUND: 1 EQUATOR? FIXED POINTS c d 0 lambda1 lambda2 lambda1+lambda2 0/1 1 4 Yes Yes No No NUMBER OF EQUATORS FOUND: 1 1 0 0 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 either one of the above cycles or the nonslope. If the slope function maps slope p/q to slope p'/q', then |q'| <= |q| for every slope p/q with |p| <= 50 and |q| <= 50. FUNDAMENTAL GROUP WREATH RECURSIONS When the translation term of the affine map is 0: NewSphereMachine( "a=<1,a*b,b,b^-1>(2,4)", "b=(1,4)(2,3)", "c=(1,4)(2,3)", "d=(2,4)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=<1,1,b,b^-1>(1,2)(3,4)", "b=<1,a*b,b,b^-1>(2,4)", "c=(2,4)", "d=(1,2)(3,4)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,3)", "b=(1,4)(2,3)", "c=(1,4)(2,3)", "d=(1,3)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(1,4)(2,3)", "b=(2,4)", "c=<1,a*b,b,b^-1>(2,4)", "d=(1,4)(2,3)", "a*b*c*d");