These Thurston maps are NET maps for every choice of translation term. They are primitive and have degree 5. 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 10. ALL THURSTON MULTIPLIERS c/d IN UNREDUCED FORM 1/5, 1/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-50.809017,-49.690983) (-49.447214,-47.690983) (-47.447214,-45.690983) (-45.447214,-43.690983) (-43.447214,-41.690983) (-41.447214,-39.690983) (-39.447214,-37.690983) (-37.447214,-35.690983) (-35.447214,-33.690983) (-33.447214,-31.690983) (-31.447214,-29.690983) (-29.447214,-27.690983) (-27.447214,-25.690983) (-25.447214,26.309017 ) ( 26.552786,28.309017 ) ( 28.552786,30.309017 ) ( 30.552786,32.309017 ) ( 32.552786,34.309017 ) ( 34.552786,36.309017 ) ( 36.552786,38.309017 ) ( 38.552786,40.309017 ) ( 40.552786,42.309017 ) ( 42.552786,44.309017 ) ( 44.552786,46.309017 ) ( 46.552786,48.309017 ) ( 48.552786,50.309017 ) 0/1 is the slope of a Thurston obstruction with c = 1 and d = 1. These NET maps are not rational. SLOPE FUNCTION INFORMATION NUMBER OF FIXED POINTS: 1 EQUATOR? FIXED POINT c d 0 lambda1 lambda2 lambda1+lambda2 0/1 1 1 No No No No NUMBER OF EQUATORS: 0 0 0 0 There are no more slope function fixed points because every loop multiplier of the mod 2 slope correspondence graph is at least 1 and there can be at most one obstruction. No nontrivial cycles were found. The slope function maps every slope to a slope: no slope maps 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 one of the slopes described above. FUNDAMENTAL GROUP WREATH RECURSIONS When the translation term of the affine map is 0: NewSphereMachine( "a=(1,4)(2,3)", "b=<1,1,1,1,c*d*c^-1>(1,4)(2,3)", "c=(1,5)(2,4)", "d=(1,5)(2,4)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=(2,5)(3,4)", "b=<1,a*b,b,b^-1,b^-1>(2,5)(3,4)", "c=<1,1,1,1,c>(1,5)(2,4)", "d=(1,5)(2,4)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=<1,1,1,1,c>(1,5)(2,4)", "b=(1,5)(2,4)", "c=<1,1,1,1,c*d*c^-1>(1,4)(2,3)", "d=(1,4)(2,3)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(1,5)(2,4)", "b=<1,1,1,1,c>(1,5)(2,4)", "c=<1,a*b,b,b^-1,b^-1>(2,5)(3,4)", "d=(2,5)(3,4)", "a*b*c*d");