These Thurston maps are NET maps for every choice of translation term. They are primitive and have degree 17. 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 0/17, 1/17, 1/1, 2/1, 3/1, 4/1, 5/1, 6/1, 7/1, 8/1, 11/1, 14/1, 15/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,0.000000) ( 0.000000,0.064516) ( 0.064722,0.065644) ( 0.067910,0.068414) ( 0.074721,0.075331) ( 0.077512,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 (-0.024942,0.012486) 0/1 EXTENDED HST ( 0.063456,0.064593) 2/31 EXTENDED HST -> HST ( 0.064038,0.065204) 11/170 HST ( 0.065526,0.066031) 5/76 HST ( 0.065798,0.066234) 7/106 HST ( 0.066179,0.066355) 11/166 HST ( 0.066317,0.066432) 15/226 HST ( 0.066309,0.066518) 19/286 HST ( 0.066488,0.066559) 30/451 HST ( 0.066538,0.066589) 42/631 HST ( 0.066572,0.066610) 58/871 HST ( 0.066602,0.066729) 1/15 EXTENDED HST ( 0.066724,0.066760) 59/884 HST ( 0.066743,0.066788) 46/689 HST ( 0.066764,0.066824) 35/524 HST ( 0.066795,0.066866) 27/404 HST ( 0.066838,0.066970) 20/299 HST ( 0.066893,0.067031) 16/239 HST ( 0.066953,0.067057) 14/209 HST ( 0.066996,0.067141) 11/164 HST ( 0.067076,0.067239) 9/134 HST ( 0.067169,0.067438) 7/104 HST ( 0.067315,0.067848) 4/59 HST ( 0.067840,0.067877) 15/221 HST ( 0.067860,0.067913) 41/604 HST ( 0.067881,0.067919) 11/162 HST ( 0.068365,0.068834) 5/73 HST ( 0.068816,0.068867) 19/276 HST ( 0.068855,0.068899) 25/363 HST ( 0.068889,0.068920) 37/537 HST ( 0.068914,0.068931) 53/769 HST ( 0.068924,0.068940) 71/1030 HST ( 0.068934,0.068947) 95/1378 HST ( 0.068942,0.068952) 127/1842 HST ( 0.068948,0.068982) 2/29 EXTENDED HST ( 0.068981,0.068991) 119/1725 HST ( 0.068986,0.068999) 91/1319 HST ( 0.068992,0.069008) 71/1029 HST ( 0.069000,0.069021) 55/797 HST ( 0.069010,0.069037) 41/594 HST ( 0.069024,0.069050) 35/507 HST ( 0.069036,0.069087) 25/362 HST ( 0.069067,0.069148) 17/246 HST ( 0.069120,0.069332) 7/101 HST ( 0.069259,0.069388) 26/375 HST ( 0.069336,0.069349) 19/274 HST ( 0.069346,0.069534) 5/72 HST ( 0.069411,0.069883) 8/115 HST ( 0.069565,0.070175) 3/43 EXTENDED HST -> HST ( 0.070171,0.070180) 4/57 EXTENDED HST ( 0.070151,0.070526) 9/128 HST ( 0.070420,0.070425) 5/71 EXTENDED HST ( 0.070513,0.071429) 6/85 HST ( 0.071287,0.071579) 1/14 EXTENDED HST ( 0.071575,0.071668) 27/377 HST ( 0.071623,0.071730) 20/279 HST ( 0.071685,0.071703) 19/265 HST ( 0.071704,0.071924) 15/209 HST ( 0.071771,0.072092) 11/153 HST ( 0.071903,0.072429) 7/97 HST ( 0.072275,0.072906) 4/55 HST ( 0.072806,0.073096) 7/96 HST ( 0.073091,0.073932) 3/41 EXTENDED HST -> HST ( 0.073251,0.076554) 2/27 EXTENDED HST -> HST ( 0.076532,0.076631) 16/209 HST ( 0.076586,0.076703) 21/274 HST ( 0.076667,0.077174) 1/13 EXTENDED HST ( 0.077001,0.077518) 19/246 HST ( 0.077251,0.077527) 12/155 HST ( 0.077419,0.077519) 11/142 HST The supplemental half-space computation shows that these NET maps are rational. SLOPE FUNCTION INFORMATION NUMBER OF FIXED POINTS: 10 EQUATOR? FIXED POINT c d 0 lambda1 lambda2 lambda1+lambda2 -14/1 1 17 Yes Yes No No 0/1 1 17 Yes Yes No No -32/3 1 17 Yes Yes No No -130/9 1 17 Yes Yes No No -128/9 1 17 Yes Yes No No -96/7 1 17 Yes Yes No No -224/15 1 17 Yes Yes No No -192/13 1 17 Yes Yes No No -160/11 1 17 Yes Yes No No -64/5 1 17 Yes Yes No No NUMBER OF EQUATORS: 10 10 0 0 There are no more slope function fixed points. Number of excluded intervals computed by the fixed point finder: 2340 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^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1>(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)", "b=(1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)", "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*d>(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)", "d=<1,1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c,c,c,c,c,c,c,c>(1,2)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=(1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)", "b=(1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)", "c=(1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)", "d=<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*d>(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)", "b=<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*d>(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)", "c=(1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)", "d=(1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)", "b=(1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)", "c=(1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)", "d=(1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,17)", "a*b*c*d");