These Thurston maps are NET maps for every choice of translation term. They are primitive and have degree 39. 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 0/3, 0/13, 0/39, 1/39, 1/13, 1/3, 2/3, 1/1, 3/3, 5/3, 2/1, 6/3, 3/1, 9/3 4/1, 5/1, 6/1, 7/1, 8/1, 9/1, 11/1, 12/1, 14/1, 16/1, 17/1, 18/1, 19/1, 23/1 31/1, 35/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,-0.016691) ( 0.017497,0.121930 ) ( 0.125000,0.485758 ) ( 0.504849,0.518591 ) ( 0.519660,0.982503 ) ( 1.017497,69.573593) The half-space computation does not determine rationality. EXCLUDED INTERVALS FOR JUST THE SUPPLEMENTAL HALF-SPACE COMPUTATION INTERVAL COMPUTED FOR HST OR EXTENDED HST (-195.347636,-111.819031) -142/1 HST (-152.298574,-89.201426 ) -111/1 HST (-119.855063,-69.978270 ) -89/1 HST ( -97.249512,-54.583822 ) -69/1 HST (-125.399346,-33.600654 ) -54/1 HST ( -45.130082,-27.703251 ) -33/1 EXTENDED HST -> HST ( -61.150635,-17.849365 ) -27/1 EXTENDED HST -> HST ( -21.756939,-12.743061 ) -17/1 EXTENDED HST -> HST ( -16.454163,-11.045837 ) -12/1 EXTENDED HST -> HST ( -12.686572,-8.480095 ) -11/1 EXTENDED HST -> HST ( -8.918980,-5.914354 ) -8/1 EXTENDED HST -> HST ( -6.205719,-5.544281 ) -47/8 EXTENDED HST -> HST ( -5.860021,-5.806645 ) -35/6 EXTENDED HST -> HST ( -5.826266,-5.610840 ) -17/3 EXTENDED HST -> HST ( -5.570838,-5.501385 ) -72/13 HST ( -5.525783,-5.140883 ) -11/2 EXTENDED HST -> HST ( -5.151388,-3.348612 ) -5/1 EXTENDED HST -> HST ( -infinity,-0.500000 ) -3/1 EXTENDED HST -> HST ( -0.690658,-0.361974 ) -1/2 EXTENDED HST -> HST ( -0.482219,-0.326423 ) -1/3 EXTENDED HST -> HST ( -0.411711,-0.232074 ) -7/22 EXTENDED HST -> HST ( -0.316013,-0.283987 ) -2/7 EXTENDED HST -> HST ( -0.252317,-0.247698 ) -1/4 EXTENDED HST ( -0.243566,-0.227566 ) -3/13 EXTENDED HST -> HST ( -0.238016,-0.216838 ) -5/22 EXTENDED HST -> HST ( -0.225723,-0.202848 ) -2/9 EXTENDED HST -> HST ( -0.204257,-0.132886 ) -1/5 EXTENDED HST -> HST ( -0.145016,-0.104984 ) -1/8 EXTENDED HST -> HST ( -0.107143,-0.100000 ) -3/29 EXTENDED HST -> HST ( -0.100616,-0.099384 ) -1/10 EXTENDED HST ( -0.099842,-0.098571 ) -13/131 HST ( -0.099233,-0.093074 ) -3/31 EXTENDED HST -> HST ( -0.096677,-0.069989 ) -1/11 EXTENDED HST -> HST ( -0.079288,-0.058839 ) -1/15 EXTENDED HST -> HST ( -0.065619,-0.045844 ) -1/17 EXTENDED HST -> HST ( -0.048992,-0.034534 ) -1/22 EXTENDED HST -> HST ( -0.038671,-0.027996 ) -1/29 EXTENDED HST -> HST ( -0.032226,-0.023330 ) -1/36 EXTENDED HST -> HST ( -0.023407,-0.023107 ) -1/43 EXTENDED HST -> HST ( -0.026560,-0.018950 ) -1/44 EXTENDED HST -> HST ( -0.019076,-0.018665 ) -1/53 HST ( -0.021484,-0.015553 ) -1/54 HST ( -0.017421,-0.013122 ) -1/65 HST ( -0.014873,-0.010768 ) -1/77 HST ( -0.012328,-0.009298 ) -1/93 HST ( -0.010742,-0.007777 ) -1/108 HST ( -0.009086,-0.006545 ) -1/129 HST ( -0.007632,-0.005525 ) -1/153 HST ( -0.005533,-0.005516 ) -1/181 HST ( -0.006386,-0.004607 ) -1/182 HST ( -0.005322,-0.003853 ) -1/218 HST ( -0.004140,0.004072 ) 0/1 EXTENDED HST ( 0.003447,0.004748 ) 1/243 HST ( 0.004117,0.005687 ) 1/205 HST ( 0.004883,0.006745 ) 1/172 HST ( 0.005832,0.008056 ) 1/144 HST ( 0.006979,0.007007 ) 1/143 HST ( 0.006999,0.009668 ) 1/120 HST ( 0.008399,0.011601 ) 1/100 HST ( 0.010037,0.013759 ) 1/84 HST ( 0.011998,0.016573 ) 1/70 HST ( 0.014481,0.020002 ) 1/58 HST ( 0.118338,0.126248 ) 5/41 EXTENDED HST -> HST ( 0.124230,0.125770 ) 1/8 EXTENDED HST ( 0.484863,0.488855 ) 18/37 EXTENDED HST -> HST ( 0.486817,0.490456 ) 22/45 EXTENDED HST -> HST ( 0.489258,0.492223 ) 26/53 HST ( 0.491125,0.493698 ) 32/65 HST ( 0.493671,0.506494 ) 1/2 EXTENDED HST ( 0.515842,0.522738 ) 41/79 HST ( 0.519113,0.519350 ) 27/52 HST ( 0.981044,0.984673 ) 57/58 HST ( 0.982422,0.987275 ) 65/66 HST ( 0.985127,0.989232 ) 78/79 HST ( 0.987342,1.012987 ) 1/1 EXTENDED HST ( 1.011301,1.015746 ) 76/75 HST ( 1.013408,1.018907 ) 64/63 HST ( 1.016066,1.018602 ) 59/58 HST ( 55.583822,98.249512 ) 70/1 HST ( 76.504202,135.995798 ) 99/1 HST ( 106.819031,190.347636 ) 136/1 HST -142.616211)(144.617188 infinity EXTENDED HST The supplemental half-space computation shows that these NET maps are rational. SLOPE FUNCTION INFORMATION NUMBER OF FIXED POINTS FOUND: 2 EQUATOR? FIXED POINT c d 0 lambda1 lambda2 lambda1+lambda2 0/1 1 39 Yes Yes No No -18/19 1 39 Yes Yes No No NUMBER OF EQUATORS FOUND: 2 2 0 0 The fixed point finder is unable to determine whether there are any more slope function fixed points. Number of excluded intervals computed by the fixed point finder: 20387 Here is their union. There are no more slope function fixed points whose negative reciprocals lie in any of the following intervals. EXCLUDED INTERVALS FOR THE FIXED POINT COMPUTATION (-infinity,0.947294) ( 0.947294,0.947294) ( 0.947294,0.947294) ( 0.947294,0.947294) ( 0.947294,infinity) 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=(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=(1,39)(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=<1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,c>(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,d,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(1,2)(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"); When the translation term of the affine map is lambda1: 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=(1,38)(2,37)(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)", "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=<1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,c>(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=(1,39)(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=<1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,c>(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=(1,38)(2,37)(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+lambda2: NewSphereMachine( "a=<1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,c,1>(1,38)(2,37)(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)", "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=(1,38)(2,37)(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)", "d=<1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,b^-1*a*b,c*d*b>(1,37)(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)(38,39)", "a*b*c*d"); ****************************INTEGER OVERFLOW REPORT***************************** Imminent integer overflow occurred during the computation of a fixed point excluded interval about 71431/148652. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 411032/433901. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 411032/433901. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 411032/433901. Imminent integer overflow halted evaluation of the slope function at slope -6832162/6472069 while computing a fixed point excluded interval about 3424368/3614893. Imminent integer overflow halted evaluation of the slope function at slope -15669520/14843649 during the search for all slope function fixed points. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 70078/73977. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 70078/73977. Imminent integer overflow halted evaluation of the slope function at slope -3457699/3275459 during the search for all slope function fixed points. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 37061/39123. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 37061/39123. Imminent integer overflow halted evaluation of the slope function at slope -1635368/1549175 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -1638214/1551871 during the search for all slope function fixed points.