These Thurston maps are NET maps for every choice of translation term. They are primitive and have degree 28. 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 0/28, 1/28, 1/14, 1/7, 1/4, 2/7, 1/2, 2/4, 3/4, 1/1, 4/4, 3/2, 5/2, 3/1 5/1, 11/2, 9/1, 11/1, 13/1, 19/1, 25/1, 26/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,0.000000) ( 0.000000,0.037736) ( 0.041855,0.044513) ( 0.044807,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 (-0.010897,0.008491) 0/1 EXTENDED HST ( 0.037684,0.038035) 2/53 HST ( 0.037952,0.038131) 7/184 HST ( 0.038058,0.038278) 4/105 HST ( 0.038263,0.038305) 25/653 HST ( 0.038286,0.038290) 17/444 HST ( 0.038291,0.038304) 9/235 HST ( 0.038296,0.038361) 10/261 HST ( 0.038345,0.038376) 15/391 HST ( 0.038362,0.038405) 18/469 HST ( 0.038392,0.038422) 27/703 HST ( 0.038414,0.038434) 38/989 HST ( 0.038423,0.038444) 53/1379 HST ( 0.038441,0.038481) 1/26 EXTENDED HST ( 0.038476,0.038486) 77/2001 HST ( 0.038481,0.038494) 56/1455 HST ( 0.038488,0.038519) 35/909 HST ( 0.038504,0.038506) 34/883 HST ( 0.038505,0.038536) 24/623 HST ( 0.038525,0.038527) 23/597 HST ( 0.038526,0.038570) 16/415 HST ( 0.038559,0.038625) 12/311 HST ( 0.038594,0.038654) 8/207 HST ( 0.038629,0.038689) 15/388 HST ( 0.038663,0.038686) 7/181 HST ( 0.038674,0.038760) 6/155 HST ( 0.038721,0.038787) 5/129 HST ( 0.038748,0.038833) 9/232 HST ( 0.038832,0.038838) 4/103 EXTENDED HST ( 0.038818,0.038939) 11/283 HST ( 0.038876,0.039012) 3/77 HST ( 0.039007,0.039130) 5/128 HST ( 0.039116,0.039145) 9/230 HST ( 0.039129,0.039167) 11/281 HST ( 0.039157,0.039185) 17/434 HST ( 0.039177,0.039192) 25/638 HST ( 0.039179,0.039205) 33/842 HST ( 0.039202,0.039207) 69/1760 HST ( 0.039204,0.039209) 87/2219 HST ( 0.039208,0.039211) 121/3086 HST ( 0.039210,0.039221) 2/51 EXTENDED HST ( 0.039220,0.039222) 139/3544 HST ( 0.039221,0.039224) 111/2830 HST ( 0.039223,0.039228) 81/2065 HST ( 0.039225,0.039233) 57/1453 HST ( 0.039230,0.039239) 41/1045 HST ( 0.039235,0.039246) 31/790 HST ( 0.039242,0.039243) 29/739 HST ( 0.039243,0.039351) 9/229 HST ( 0.039319,0.039414) 5/127 HST ( 0.039395,0.039464) 11/279 HST ( 0.039461,0.039467) 53/1343 HST ( 0.039466,0.039469) 83/2103 HST ( 0.039466,0.039472) 110/2787 HST ( 0.039469,0.039478) 3/76 EXTENDED HST ( 0.039477,0.039484) 79/2001 HST ( 0.039480,0.039484) 64/1621 HST ( 0.039482,0.039498) 31/785 HST ( 0.039491,0.039511) 19/481 HST ( 0.039504,0.039523) 13/329 HST ( 0.039520,0.039663) 4/101 HST ( 0.039657,0.039765) 5/126 HST ( 0.039737,0.039810) 7/176 HST ( 0.039794,0.039853) 9/226 HST ( 0.039843,0.039866) 11/276 HST ( 0.039864,0.039893) 12/301 HST ( 0.039853,0.039924) 15/376 HST ( 0.039919,0.039942) 22/551 HST ( 0.039936,0.039954) 28/701 HST ( 0.039947,0.040057) 1/25 EXTENDED HST ( 0.040052,0.040077) 25/624 HST ( 0.040065,0.040069) 24/599 HST ( 0.040069,0.040099) 20/499 HST ( 0.040084,0.040132) 14/349 HST ( 0.040120,0.040171) 11/274 HST ( 0.040148,0.040208) 9/224 HST ( 0.040191,0.040266) 7/174 HST ( 0.040252,0.040399) 5/124 HST ( 0.040355,0.040583) 4/99 HST ( 0.040418,0.040764) 3/74 HST ( 0.040677,0.040842) 17/417 HST ( 0.040804,0.040829) 2/49 EXTENDED HST ( 0.040778,0.041107) 7/171 HST ( 0.040970,0.041259) 3/73 HST ( 0.040997,0.041964) 5/121 HST ( 0.041625,0.041711) 1/24 EXTENDED HST ( 0.043278,0.048796) 3/67 HST The supplemental half-space computation shows that these NET maps are rational. SLOPE FUNCTION INFORMATION NUMBER OF FIXED POINTS FOUND: 17 EQUATOR? FIXED POINT c d 0 lambda1 lambda2 lambda1+lambda2 -26/1 1 28 Yes Yes No No -25/1 1 14 No No No No -24/1 1 28 Yes Yes No No -18/1 1 28 Yes Yes No No 0/1 1 28 Yes Yes No No -76/3 1 28 Yes Yes No No -180/7 1 28 Yes Yes No No -434/17 1 28 Yes Yes No No -126/5 1 28 Yes Yes No No -324/13 1 28 Yes Yes No No -648/25 1 28 Yes Yes No No -594/23 1 28 Yes Yes No No -486/19 1 28 Yes Yes No No -432/17 1 28 Yes Yes No No -270/11 1 28 Yes Yes No No -162/7 1 28 Yes Yes No No -108/5 1 28 Yes Yes No No NUMBER OF EQUATORS FOUND: 16 16 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: 5731 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,0.039171) ( 0.039171,0.039171) ( 0.039171,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 |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,b,b,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1>(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)", "b=(1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)", "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^-1,c^-1,c,c,c,c,c,c,c,c,c,c,c,c,c,c*d>(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)", "d=<1,1,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^-1,c,c,c,c,c,c,c,c,c,c,c,c,c>(1,2)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=(1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)", "b=(1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,15)", "c=(1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)", "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^-1,c^-1,c,c,c,c,c,c,c,c,c,c,c,c,c,c*d>(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)", "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^-1,c^-1,c,c,c,c,c,c,c,c,c,c,c,c,c,c*d>(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)", "c=(1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)", "d=(1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,15)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,15)", "b=(1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)", "c=(1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,15)", "d=(1,26)(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,28)", "a*b*c*d"); ****************************INTEGER OVERFLOW REPORT***************************** Imminent integer overflow halted evaluation of the slope function at slope -6486929/254095 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -8505294/333155 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -5610397/219761 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -2673504/104723 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -6845654/268149 during the search for all slope function fixed points.