These Thurston maps are NET maps for every choice of translation term. They are primitive and have degree 23. 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 0/23, 1/23, 1/1, 2/1, 3/1, 4/1, 5/1, 6/1, 7/1, 8/1, 9/1, 10/1, 11/1, 14/1 17/1, 20/1, 21/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,0.000000) ( 0.000000,0.046512) ( 0.054511,0.057231) ( 0.059230,infinity) The half-space computation does not determine rationality. EXCLUDED INTERVALS FOR JUST THE SUPPLEMENTAL HALF-SPACE COMPUTATION INTERVAL COMPUTED FOR HST OR EXTENDED HST (-0.014640,0.008045) 0/1 EXTENDED HST ( 0.046012,0.046520) 2/43 EXTENDED HST -> HST ( 0.046509,0.046624) 19/408 HST ( 0.046571,0.046757) 9/193 HST ( 0.046657,0.047066) 3/64 HST ( 0.047057,0.047145) 17/361 HST ( 0.047099,0.047247) 5/106 HST ( 0.047242,0.047356) 7/148 HST ( 0.047277,0.047441) 9/190 HST ( 0.047412,0.047479) 13/274 HST ( 0.047460,0.047512) 17/358 HST ( 0.047495,0.047529) 22/463 HST ( 0.047512,0.047556) 26/547 HST ( 0.047545,0.047571) 36/757 HST ( 0.047563,0.047583) 48/1009 HST ( 0.047576,0.047590) 63/1324 HST ( 0.047585,0.047597) 80/1681 HST ( 0.047593,0.047602) 104/2185 HST ( 0.047598,0.047639) 1/21 EXTENDED HST ( 0.047638,0.047648) 93/1952 HST ( 0.047644,0.047661) 70/1469 HST ( 0.047652,0.047669) 55/1154 HST ( 0.047661,0.047683) 44/923 HST ( 0.047671,0.047697) 35/734 HST ( 0.047684,0.047723) 27/566 HST ( 0.047705,0.047749) 21/440 HST ( 0.047732,0.047771) 17/356 HST ( 0.047760,0.047827) 14/293 HST ( 0.047793,0.047856) 12/251 HST ( 0.047824,0.047919) 10/209 HST ( 0.047856,0.048053) 6/125 HST ( 0.048033,0.048111) 5/104 HST ( 0.048101,0.048133) 23/478 HST ( 0.048119,0.048176) 9/187 HST ( 0.048170,0.048331) 4/83 HST ( 0.048279,0.048420) 16/331 HST ( 0.048380,0.048394) 3/62 EXTENDED HST ( 0.048294,0.048510) 14/289 HST ( 0.048453,0.048485) 11/227 HST ( 0.048484,0.048530) 8/165 HST ( 0.048489,0.048635) 5/103 HST ( 0.048570,0.048650) 9/185 HST ( 0.048648,0.048667) 38/781 HST ( 0.048657,0.048690) 11/226 HST ( 0.048686,0.048717) 15/308 HST ( 0.048704,0.048732) 19/390 HST ( 0.048727,0.048746) 25/513 HST ( 0.048740,0.048753) 35/718 HST ( 0.048748,0.048762) 45/923 HST ( 0.048758,0.048765) 63/1292 HST ( 0.048762,0.048769) 79/1620 HST ( 0.048767,0.048771) 105/2153 HST ( 0.048770,0.048773) 133/2727 HST ( 0.048772,0.048789) 2/41 EXTENDED HST ( 0.048780,0.048820) 63/1291 HST ( 0.048799,0.048810) 49/1004 HST ( 0.048806,0.048820) 39/799 HST ( 0.048812,0.048831) 29/594 HST ( 0.048824,0.048851) 21/430 HST ( 0.048839,0.048903) 13/266 HST ( 0.048888,0.048937) 9/184 HST ( 0.048936,0.048948) 30/613 HST ( 0.048946,0.048950) 100/2043 HST ( 0.048948,0.049111) 5/102 HST ( 0.048980,0.049231) 14/285 HST ( 0.049173,0.049188) 3/61 EXTENDED HST ( 0.049179,0.049282) 16/325 HST ( 0.049274,0.049316) 31/629 HST ( 0.049292,0.049670) 4/81 HST ( 0.049626,0.049718) 38/765 HST ( 0.049675,0.049747) 8/161 HST ( 0.049712,0.049770) 10/201 HST ( 0.049719,0.049836) 11/221 HST ( 0.049816,0.049881) 16/321 HST ( 0.049861,0.049908) 21/421 HST ( 0.049896,0.049929) 28/561 HST ( 0.049865,0.050000) 36/721 HST ( 0.049948,0.050054) 1/20 EXTENDED HST ( 0.050000,0.050459) 9/179 HST ( 0.050299,0.050704) 6/119 HST ( 0.050454,0.050665) 5/99 HST ( 0.050576,0.051121) 3/59 HST ( 0.051116,0.052681) 2/39 EXTENDED HST -> HST ( 0.052595,0.054740) 1/19 EXTENDED HST -> HST ( 0.055474,0.060472) 3/52 HST ( 0.058730,0.058919) 1/17 EXTENDED HST The supplemental half-space computation shows that these NET maps are rational. SLOPE FUNCTION INFORMATION NUMBER OF FIXED POINTS: 13 EQUATOR? FIXED POINT c d 0 lambda1 lambda2 lambda1+lambda2 -20/1 1 23 Yes Yes No No 0/1 1 23 Yes Yes No No -44/3 1 23 Yes Yes No No -62/3 1 23 Yes Yes No No -440/21 1 23 Yes Yes No No -308/15 1 23 Yes Yes No No -266/13 1 23 Yes Yes No No -264/13 1 23 Yes Yes No No -396/19 1 23 Yes Yes No No -352/17 1 23 Yes Yes No No -176/9 1 23 Yes Yes No No -132/7 1 23 Yes Yes No No -88/5 1 23 Yes Yes No No NUMBER OF EQUATORS: 13 13 0 0 There are no more slope function fixed points. Number of excluded intervals computed by the fixed point finder: 3604 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,b,b,b,b,b,b,b,b,b,b,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1>(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)", "b=(1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)", "c=<1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c,c,c,c,c,c,c,c,c,c,c*d>(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)", "d=<1,1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c,c,c,c,c,c,c,c,c,c,c>(1,2)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=(1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)", "b=(1,22)(2,21)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)", "c=(1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)", "d=<1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c,c,c,c,c,c,c,c,c,c,c*d>(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)", "b=<1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c,c,c,c,c,c,c,c,c,c,c*d>(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)", "c=(1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)", "d=(1,22)(2,21)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(1,22)(2,21)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)", "b=(1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)", "c=(1,22)(2,21)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)", "d=(1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(22,23)", "a*b*c*d");