These Thurston maps are primitive and have degree 4. There are exactly three postcritical points if the translation term b = 0. For this value of b, this Thurston map is not a NET map. All other choices of b among 0, lambda1, lambda2, lambda1+lambda2 yield NET maps. PURE MODULAR GROUP HURWITZ EQUIVALENCE CLASSES FOR TRANSLATIONS {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 9. ALL THURSTON MULTIPLIERS c/d IN UNREDUCED FORM 0/2, 1/2 Since every Thurston multiplier is less than 1, every NET map in this modular group Hurwitz class is rational. EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-110.355339,-1.006102 ) ( -0.993935,-0.503070 ) ( -0.496968,-0.009439 ) ( 0.009439,0.496968 ) ( 0.503070,0.993935 ) ( 1.006102,110.355339) SLOPE FUNCTION INFORMATION NUMBER OF FIXED POINTS: 2 EQUATOR? FIXED POINT c d 0 lambda1 lambda2 lambda1+lambda2 1/0 1 2 * No No No 0/1 1 2 * No No No NUMBER OF EQUATORS: * 0 0 0 There are no more slope function fixed points. Number of excluded intervals computed by the fixed point finder: 110 There are no equators because both elementary divisors are greater than 1. 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 |p'| <= |p| for every slope p/q with |p| <= 50 and |q| <= 50. 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 lambda1: NewSphereMachine( "a=(1,3)(2,4)", "b=", "c=<1,d,1,c>(1,2)(3,4)", "d=(1,4)(2,3)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=<1,c^-1,c*d,c>(1,4)(2,3)", "b=<1,1,1,1>(1,2)(3,4)", "c=(1,3)(2,4)", "d=<1,1,b^-1*a*b,1>", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(1,4)(2,3)", "b=<1,d,1,c>(1,2)(3,4)", "c=", "d=(1,3)(2,4)", "a*b*c*d");