These Thurston maps are NET maps for every choice of translation term. They are primitive and have degree 8. 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 3. ALL THURSTON MULTIPLIERS c/d IN UNREDUCED FORM 0/4, 1/8, 2/2, 2/1, 6/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,-0.008673) ( 0.008673,infinity ) 1/0 is the slope of a Thurston obstruction with c = 6 and d = 1. These NET maps are not rational. SLOPE FUNCTION INFORMATION NUMBER OF FIXED POINTS: 2 EQUATOR? FIXED POINT c d 0 lambda1 lambda2 lambda1+lambda2 1/0 6 1 No No No No 0/1 1 8 Yes Yes No No NUMBER OF EQUATORS: 1 1 0 0 There are no more slope function fixed points. Number of excluded intervals computed by the fixed point finder: 9 No nontrivial cycles were found. Here is the action of the slope function on an invariant set S of slopes. N/1 -> 6N/1 The set S contains infinitely many infinite slope function trajectory tails. 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 1/0 is n, then the intersection pairing of s' with 1/0 is at most n. The slope function orbit of every slope whose intersection pairing pairing with 1/0 is at most 50 either ends in the nonslope or ends in one of the slopes described above or it has an infinite tail in one of the infinite sets described above. FUNDAMENTAL GROUP WREATH RECURSIONS When the translation term of the affine map is 0: NewSphereMachine( "a=<1,b,b,b,1,b^-1,b^-1,b^-1>(2,8)(3,7)(4,6)", "b=(1,8)(2,7)(3,6)(4,5)", "c=(1,8)(2,7)(3,6)(4,5)", "d=<1,c^-1,c^-1,c^-1,1,c,c,c>(2,8)(3,7)(4,6)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=(1,2)(3,8)(4,7)(5,6)", "b=<1,b,b,b,1,b^-1,b^-1,b^-1>(2,8)(3,7)(4,6)", "c=<1,c^-1,c^-1,c^-1,1,c,c,c>(2,8)(3,7)(4,6)", "d=<1,d,c^-1,c^-1,1,c,c,c>(1,2)(3,8)(4,7)(5,6)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,7)(2,6)(3,5)", "b=(1,8)(2,7)(3,6)(4,5)", "c=(1,8)(2,7)(3,6)(4,5)", "d=(1,7)(2,6)(3,5)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(1,8)(2,7)(3,6)(4,5)", "b=<1,c^-1,c^-1,c^-1,1,c,c,c>(2,8)(3,7)(4,6)", "c=<1,b,b,b,1,b^-1,b^-1,b^-1>(2,8)(3,7)(4,6)", "d=(1,8)(2,7)(3,6)(4,5)", "a*b*c*d");