These Thurston maps are NET maps for every choice of translation term. They have degree 16. They are imprimitive, each factoring as a NET map with degree 8 followed by a Euclidean NET map with degree 2. PURE MODULAR GROUP HURWITZ EQUIVALENCE CLASSES FOR TRANSLATIONS {0} {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 7. ALL THURSTON MULTIPLIERS c/d IN UNREDUCED FORM 0/2, 0/4, 0/16, 2/8, 2/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,-0.033784) ( 0.034247,infinity ) 1/0 is the slope of a Thurston obstruction with c = 2 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 1/0 2 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 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 with 1/0 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=<1,a*b,b,b,b,c^-1*b,b,b,1,1,b^-1,b^-1*c,b^-1*c,b^-1,b^-1,b^-1>(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)", "b=(1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)", "c=(1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)", "d=<1,a*b,a*b,c^-1,c^-1,c^-1,1,1,1,1,1,c,c,c*d,c*d,c*d>(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=<1,1,b,b,b,c^-1*b,b,b,1,1,b^-1,b^-1,b^-1*c,b^-1,b^-1,b^-1>(1,2)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)", "b=<1,a*b,b,b,b,c^-1*b,b,b,1,1,b^-1,b^-1*c,b^-1*c,b^-1,b^-1,b^-1>(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)", "c=<1,a*b,a*b,c^-1,c^-1,c^-1,1,1,1,1,1,c,c,c*d,c*d,c*d>(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)", "d=<1,1,a*b,a*b,c^-1,c^-1,1,1,1,1,1,1,c,c,c*d,c*d>(1,2)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)", "b=(1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)", "c=(1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)", "d=(1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)", "b=<1,a*b,a*b,c^-1,c^-1,c^-1,1,1,1,1,1,c,c,c*d,c*d,c*d>(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)", "c=<1,a*b,b,b,b,c^-1*b,b,b,1,1,b^-1,b^-1*c,b^-1*c,b^-1,b^-1,b^-1>(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)", "d=(1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)", "a*b*c*d");