These Thurston maps are NET maps for every choice of translation term. They are primitive and have degree 40. 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 24. ALL THURSTON MULTIPLIERS c/d IN UNREDUCED FORM 0/1, 0/5, 0/8, 0/10, 0/40, 1/40, 1/20, 1/10, 1/8, 1/5, 1/4, 2/5, 1/2, 2/4 1/1, 2/2, 4/4, 2/1, 5/2, 6/2, 7/2, 4/1, 5/1, 6/1, 7/1, 14/2, 9/1, 11/1, 12/1 13/1, 14/1, 15/1, 17/1, 19/1, 22/1, 34/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-59.390456,-1.017126) ( -0.982763,-0.013928) ( 0.013719,59.390456) The half-space computation does not determine rationality. EXCLUDED INTERVALS FOR JUST THE SUPPLEMENTAL HALF-SPACE COMPUTATION INTERVAL COMPUTED FOR HST OR EXTENDED HST (-1.019577,-1.014308) -60/59 HST (-1.014431,-1.014143) -71/70 HST (-1.016348,-1.011831) -72/71 HST (-1.012199,-1.011442) -428/423 HST (-1.011814,-1.011716) -86/85 HST (-1.011446,-1.011283) -89/88 HST (-1.013036,-1.009443) -90/89 HST (-1.009454,-1.009414) -107/106 HST (-1.010840,-1.007856) -108/107 HST (-1.008989,-1.006518) -129/128 HST (-1.007577,-1.005497) -309/307 HST (-1.006503,-1.006484) -155/154 HST (-1.005501,-1.005488) -183/182 HST (-1.006334,-1.004596) -184/183 HST (-1.004837,-1.004351) -656/653 HST (-1.004592,-1.004582) -219/218 HST (-1.004361,-1.004335) -231/230 HST (-1.005017,-1.003642) -232/231 HST (-1.003969,-0.996063) -1/1 EXTENDED HST (-0.996147,-0.994723) -218/219 HST (-0.995418,-0.995408) -217/218 HST (-0.995396,-0.993677) -182/183 HST (-0.994512,-0.994499) -181/182 HST (-0.994492,-0.992438) -152/153 HST (-0.993448,-0.993394) -151/152 HST (-0.993379,-0.990870) -126/127 HST (-0.992120,-0.989192) -106/107 HST (-0.990586,-0.990546) -105/106 HST (-0.990508,-0.990444) -104/105 HST (-0.990443,-0.990326) -103/104 HST (-0.990306,-0.986712) -86/87 HST (-0.988402,-0.988342) -85/86 HST (-0.988284,-0.988186) -84/85 HST (-0.988116,-0.983725) -70/71 HST (-0.985857,-0.985569) -69/70 HST (-0.985615,-0.980528) -58/59 HST (-0.014012,-0.013768) -1/72 HST (-0.016485,-0.010524) -1/73 HST (-0.014039,-0.006908) -1/96 HST (-0.009245,-0.004502) -1/145 HST (-0.005468,0.005528 ) 0/1 EXTENDED HST ( 0.005359,0.008317 ) 1/146 HST ( 0.006856,0.014143 ) 1/96 HST ( 0.010460,0.016585 ) 1/73 HST -9.943359)(7.944336 infinity EXTENDED HST The supplemental half-space computation shows that these NET maps are rational. SLOPE FUNCTION INFORMATION NUMBER OF FIXED POINTS: 1 EQUATOR? FIXED POINT c d 0 lambda1 lambda2 lambda1+lambda2 1/0 1 40 No Yes No Yes NUMBER OF EQUATORS: 0 1 0 1 There are no more slope function fixed points. Number of excluded intervals computed by the fixed point finder: 10005 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 0: NewSphereMachine( "a=(1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)", "b=(1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)", "c=(1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)", "d=(1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=<1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,c,c,c,c,c,c,c,c,c,c*d>(2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)", "b=<1,b,c^-1*b,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,c,c,c,c,c,c,c,c,c,b^-1*c>(2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)", "c=(1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)", "d=(1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)", "b=(1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)", "c=(1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)", "d=(1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)", "b=(1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)", "c=<1,b,c^-1*b,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,c,c,c,c,c,c,c,c,c,b^-1*c>(2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)", "d=<1,c^-1*a^-1*b,c^-1*b,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,c,c,c,c,c,c,c,c,b^-1*c,b^-1*d*a*c>(2,40)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)", "a*b*c*d");