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,lambda2} {lambda1,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 6. ALL THURSTON MULTIPLIERS c/d IN UNREDUCED FORM 0/2, 1/2, 2/2 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-24.666667,-24.000000) (-24.000000,-23.333333) (-22.666667,-22.000000) (-22.000000,-21.333333) (-20.666667,-20.000000) (-20.000000,-19.333333) (-18.666667,-18.000000) (-18.000000,-17.333333) (-16.920991,-16.000000) (-16.000000,-15.079009) (-14.920991,-14.000000) (-14.000000,-13.079009) (-12.920991,-12.000000) (-12.000000,-11.000000) (-11.000000,-10.000000) (-10.000000,-9.000000 ) ( -9.000000,-8.000000 ) ( -8.000000,-7.000000 ) ( -7.000000,-6.000000 ) ( -6.000000,-5.000000 ) ( -5.000000,-4.000000 ) ( -4.000000,-3.000000 ) ( -3.000000,-2.000000 ) ( -2.000000,-1.000000 ) ( -1.000000,0.000000 ) ( 0.000000,1.000000 ) ( 1.000000,2.000000 ) ( 2.000000,3.000000 ) ( 3.000000,4.000000 ) ( 4.000000,5.000000 ) ( 5.000000,6.000000 ) ( 6.000000,7.000000 ) ( 7.000000,8.000000 ) ( 8.000000,9.000000 ) ( 9.000000,10.000000 ) ( 10.000000,11.000000 ) ( 11.000000,12.000000 ) ( 12.000000,12.920991 ) ( 13.079009,14.000000 ) ( 14.000000,14.920991 ) ( 15.079009,16.000000 ) ( 16.000000,16.920991 ) ( 17.333333,18.000000 ) ( 18.000000,18.666667 ) ( 19.333333,20.000000 ) ( 20.000000,20.666667 ) ( 21.333333,22.000000 ) ( 22.000000,22.666667 ) ( 23.333333,24.000000 ) ( 24.000000,24.666667 ) 0/1 is the slope of a Thurston obstruction with c = 2 and d = 2. These NET maps are not rational. SLOPE FUNCTION INFORMATION EQUATOR? FIXED POINT c d 0 lambda1 lambda2 lambda1+lambda2 0/1 2 2 No No No No 1/2N 1 2 No No No No There are no more slope function fixed points. Number of excluded intervals computed by the fixed point finder: 54 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. If the slope function maps slope s to a slope s' and if the intersection pairing of s with 0/1 is n, then the intersection pairing of s' with 0/1 is at most n. The slope function orbit of every slope whose intersection pairing with 0/1 is at most 50 ends in either the nonslope or one of the slopes described above. FUNDAMENTAL GROUP WREATH RECURSIONS When the translation term of the affine map is 0: NewSphereMachine( "a=", "b=(1,3)(2,4)", "c=<1,1,1,1>(1,2)(3,4)", "d=(1,4)(2,3)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=(1,3)(2,4)", "b=", "c=<1,1,1,1>(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,a*b,c*d,1>(1,4)(2,3)", "b=<1,1,1,1>(1,2)(3,4)", "c=(1,3)(2,4)", "d=<1,d,b^-1*a*b,1>", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=<1,a*b,c*d,1>(1,4)(2,3)", "b=<1,1,1,1>(1,2)(3,4)", "c=", "d=(1,3)(2,4)", "a*b*c*d");