These Thurston maps are NET maps for every choice of translation term. They are primitive and have degree 36. 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 24. ALL THURSTON MULTIPLIERS c/d IN UNREDUCED FORM 0/12, 0/18, 0/36, 1/36, 1/9, 1/6, 1/4, 1/3, 1/2, 2/4, 3/4, 1/1, 3/3, 4/3 3/2, 5/3, 5/2, 3/1, 6/2, 4/1, 5/1, 11/2, 6/1, 7/1, 13/1, 29/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,-0.016667 ) ( 0.017483,0.492308 ) ( 0.507353,111.203194) The half-space computation does not determine rationality. EXCLUDED INTERVALS FOR JUST THE SUPPLEMENTAL HALF-SPACE COMPUTATION INTERVAL COMPUTED FOR HST OR EXTENDED HST (-114.385165,-103.614835) -109/1 HST (-169.230203,-75.169797 ) -103/1 HST ( -infinity,-37.500000 ) -75/1 HST ( -42.385165,-31.614835 ) -37/1 EXTENDED HST -> HST ( -48.937347,-23.462653 ) -31/1 EXTENDED HST -> HST ( -40.683429,-16.701187 ) -23/1 EXTENDED HST -> HST ( -23.500000,-13.750000 ) -16/1 EXTENDED HST -> HST ( -14.290994,-11.709006 ) -13/1 EXTENDED HST -> HST ( -12.500000,-7.250000 ) -11/1 EXTENDED HST -> HST ( -11.139388,-5.260612 ) -7/1 EXTENDED HST -> HST ( -7.000000,-4.000000 ) -5/1 EXTENDED HST -> HST ( -5.300000,-3.928571 ) -4/1 EXTENDED HST -> HST ( -infinity,-0.500000 ) -3/1 EXTENDED HST -> HST ( -0.700000,-0.357143 ) -1/2 EXTENDED HST -> HST ( -0.376141,-0.339972 ) -5/14 EXTENDED HST -> HST ( -0.344311,-0.334320 ) -18/53 HST ( -0.338509,-0.328221 ) -1/3 EXTENDED HST ( -0.364198,-0.293785 ) -19/58 HST ( -0.327586,-0.274194 ) -2/7 EXTENDED HST -> HST ( -0.312500,-0.235294 ) -3/11 EXTENDED HST -> HST ( -0.252613,-0.247405 ) -1/4 EXTENDED HST ( -0.250000,-0.225806 ) -4/17 EXTENDED HST -> HST ( -0.231707,-0.197674 ) -2/9 EXTENDED HST -> HST ( -0.214286,-0.187500 ) -1/5 EXTENDED HST -> HST ( -0.194444,-0.138889 ) -1/6 EXTENDED HST -> HST ( -0.145833,-0.104167 ) -1/8 EXTENDED HST -> HST ( -0.105263,-0.095238 ) -1/10 EXTENDED HST -> HST ( -0.100000,-0.067568 ) -1/11 EXTENDED HST -> HST ( -0.083333,-0.059524 ) -1/15 EXTENDED HST -> HST ( -0.060000,-0.057692 ) -1/17 EXTENDED HST -> HST ( -0.064815,-0.046296 ) -1/18 EXTENDED HST -> HST ( -0.049296,-0.034247 ) -1/22 EXTENDED HST -> HST ( -0.039326,-0.027473 ) -1/30 EXTENDED HST -> HST ( -0.027778,-0.026316 ) -1/37 EXTENDED HST -> HST ( -0.030973,-0.021739 ) -1/38 EXTENDED HST -> HST ( -0.021978,-0.021505 ) -1/46 EXTENDED HST -> HST ( -0.024476,-0.017241 ) -1/47 EXTENDED HST -> HST ( -0.019553,-0.013812 ) -1/58 HST ( -0.013889,-0.013514 ) -1/73 HST ( -0.015837,-0.011211 ) -1/74 HST ( -0.012963,-0.009259 ) -1/90 HST ( -0.010802,-0.007716 ) -1/108 HST ( -0.008861,-0.006297 ) -1/130 HST ( -0.007384,-0.005274 ) -1/159 HST ( -0.005401,-0.003858 ) -1/214 HST ( -0.004673,0.004587 ) 0/1 EXTENDED HST ( 0.003858,0.005401 ) 1/216 HST ( 0.004630,0.006481 ) 1/180 HST ( 0.005568,0.007761 ) 1/150 HST ( 0.006693,0.006730 ) 1/149 HST ( 0.006739,0.009383 ) 1/124 HST ( 0.008108,0.008152 ) 1/123 HST ( 0.008170,0.011438 ) 1/102 HST ( 0.009861,0.009941 ) 1/101 HST ( 0.009960,0.013834 ) 1/84 HST ( 0.011962,0.016588 ) 1/70 HST ( 0.014368,0.020115 ) 1/58 HST ( 0.490099,0.495000 ) 32/65 HST ( 0.492958,0.507043 ) 1/2 EXTENDED HST ( 0.504950,0.509434 ) 35/69 HST ( 79.951634,191.648366 ) 112/1 HST -109.999023)(112.000000 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 0/1 1 36 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: 12476 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 |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,b,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,b^-1>(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)", "b=(1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)", "c=<1,c^-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,c,c*d>(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)", "d=<1,1,c^-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,c>(1,2)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=(1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)", "b=(1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)", "c=(1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)", "d=<1,c^-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,c,c*d>(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)", "b=<1,c^-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,c,c*d>(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)", "c=(1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)", "d=(1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)", "b=(1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)", "c=(1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)", "d=(1,34)(2,33)(3,32)(4,31)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(35,36)", "a*b*c*d");