These Thurston maps are NET maps for every choice of translation term. They are primitive and have degree 32. 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 0/32, 1/32, 1/16, 1/8, 1/4, 1/2, 3/4, 1/1, 5/4, 3/2, 5/2, 3/1, 7/2, 5/1 13/2, 7/1, 9/1, 11/1, 13/1, 15/1, 17/1, 23/1, 29/1, 30/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,0.000000) ( 0.000000,0.032787) ( 0.035335,0.036271) ( 0.036334,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 (-0.008850,0.006993) 0/1 EXTENDED HST ( 0.032781,0.032793) 2/61 EXTENDED HST ( 0.032691,0.032952) 11/335 HST ( 0.032836,0.033079) 3/91 HST ( 0.033063,0.033095) 13/393 HST ( 0.033081,0.033110) 51/1541 HST ( 0.033096,0.033127) 5/151 HST ( 0.033117,0.033141) 27/815 HST ( 0.033129,0.033219) 6/181 HST ( 0.033205,0.033259) 10/301 HST ( 0.033245,0.033275) 15/451 HST ( 0.033257,0.033291) 19/571 HST ( 0.033281,0.033303) 27/811 HST ( 0.033300,0.033305) 109/3273 HST ( 0.033303,0.033304) 37/1111 HST ( 0.033302,0.033309) 119/3573 HST ( 0.033305,0.033307) 40/1201 HST ( 0.033308,0.033309) 45/1351 HST ( 0.033304,0.033315) 46/1381 HST ( 0.033311,0.033320) 62/1861 HST ( 0.033317,0.033323) 84/2521 HST ( 0.033323,0.033327) 130/3901 HST ( 0.033325,0.033342) 1/30 EXTENDED HST ( 0.033341,0.033346) 108/3239 HST ( 0.033344,0.033350) 81/2429 HST ( 0.033347,0.033356) 60/1799 HST ( 0.033352,0.033352) 59/1769 HST ( 0.033352,0.033363) 44/1319 HST ( 0.033359,0.033359) 43/1289 HST ( 0.033359,0.033375) 33/989 HST ( 0.033367,0.033388) 25/749 HST ( 0.033379,0.033384) 23/689 HST ( 0.033382,0.033416) 18/539 HST ( 0.033397,0.033423) 16/479 HST ( 0.033404,0.033431) 12/359 HST ( 0.033429,0.033484) 9/269 HST ( 0.033471,0.033523) 7/209 HST ( 0.033510,0.033600) 5/149 HST ( 0.033598,0.033603) 21/625 HST ( 0.033599,0.033608) 29/863 HST ( 0.033607,0.033609) 49/1458 HST ( 0.033604,0.033614) 61/1815 HST ( 0.033611,0.033615) 4/119 EXTENDED HST ( 0.033608,0.033629) 59/1755 HST ( 0.033619,0.033684) 7/208 HST ( 0.033668,0.033759) 3/89 HST ( 0.033748,0.033813) 5/148 HST ( 0.033806,0.033829) 7/207 HST ( 0.033825,0.033845) 9/266 HST ( 0.033815,0.033882) 11/325 HST ( 0.033880,0.033885) 37/1092 HST ( 0.033883,0.033888) 45/1328 HST ( 0.033885,0.033890) 55/1623 HST ( 0.033889,0.033892) 73/2154 HST ( 0.033889,0.033895) 91/2685 HST ( 0.033894,0.033903) 2/59 EXTENDED HST ( 0.033903,0.033904) 111/3274 HST ( 0.033903,0.033906) 91/2684 HST ( 0.033905,0.033910) 65/1917 HST ( 0.033907,0.033913) 49/1445 HST ( 0.033910,0.033927) 29/855 HST ( 0.033919,0.033938) 21/619 HST ( 0.033928,0.033936) 19/560 HST ( 0.033929,0.033954) 13/383 HST ( 0.033948,0.033976) 9/265 HST ( 0.033962,0.034056) 5/147 HST ( 0.034048,0.034139) 3/88 HST ( 0.034129,0.034162) 7/205 HST ( 0.034125,0.034196) 37/1083 HST ( 0.034165,0.034259) 4/117 HST ( 0.034252,0.034287) 11/321 HST ( 0.034285,0.034287) 6/175 EXTENDED HST ( 0.034286,0.034291) 67/1954 HST ( 0.034289,0.034340) 7/204 HST ( 0.034337,0.034368) 9/262 HST ( 0.034362,0.034390) 11/320 HST ( 0.034377,0.034407) 13/378 HST ( 0.034388,0.034432) 16/465 HST ( 0.034428,0.034443) 24/697 HST ( 0.034438,0.034451) 30/871 HST ( 0.034448,0.034520) 1/29 EXTENDED HST ( 0.034515,0.034528) 31/898 HST ( 0.034522,0.034538) 26/753 HST ( 0.034529,0.034550) 22/637 HST ( 0.034538,0.034570) 16/463 HST ( 0.034561,0.034586) 14/405 HST ( 0.034573,0.034608) 11/318 HST ( 0.034599,0.034630) 9/260 HST ( 0.034620,0.034689) 8/231 HST ( 0.034650,0.034826) 4/115 HST ( 0.034810,0.034857) 31/890 HST ( 0.034833,0.034929) 3/86 HST ( 0.034909,0.035014) 5/143 HST ( 0.034965,0.035071) 9/257 HST ( 0.034997,0.035149) 37/1055 HST ( 0.035080,0.035096) 2/57 EXTENDED HST ( 0.035109,0.035634) 3/85 HST ( 0.036192,0.036377) 16/441 HST ( 0.036287,0.036320) 9/248 HST The supplemental half-space computation shows that these NET maps are rational. SLOPE FUNCTION INFORMATION NUMBER OF FIXED POINTS FOUND: 18 EQUATOR? FIXED POINT c d 0 lambda1 lambda2 lambda1+lambda2 -30/1 1 32 Yes Yes No No -29/1 1 16 No No No No 0/1 1 32 Yes Yes No No -560/19 1 32 Yes Yes No No -434/15 1 32 Yes Yes No No -248/9 1 32 Yes Yes No No -868/29 1 32 Yes Yes No No -806/27 1 32 Yes Yes No No -744/25 1 32 Yes Yes No No -682/23 1 32 Yes Yes No No -620/21 1 32 Yes Yes No No -558/19 1 32 Yes Yes No No -496/17 1 32 Yes Yes No No -372/13 1 32 Yes Yes No No -310/11 1 32 Yes Yes No No -186/7 1 32 Yes Yes No No -124/5 1 32 Yes Yes No No -62/3 1 32 Yes Yes No No NUMBER OF EQUATORS FOUND: 17 17 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: 6592 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,0.033928) ( 0.033928,0.033928) ( 0.033928,0.033928) ( 0.033928,0.033929) ( 0.033929,0.033929) ( 0.033929,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,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,b^-1,b^-1>(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)", "b=(1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)", "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^-1,c^-1,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c*d>(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)", "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^-1,c^-1,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c>(1,2)(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)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=(1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)", "b=(1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)", "c=(1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)", "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^-1,c^-1,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c*d>(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)", "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^-1,c^-1,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c*d>(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)", "c=(1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)", "d=(1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)", "b=(1,32)(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)", "c=(1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)", "d=(1,30)(2,29)(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)(31,32)", "a*b*c*d"); ****************************INTEGER OVERFLOW REPORT***************************** Imminent integer overflow halted evaluation of the slope function at slope -1812510/60449 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -2145658/72441 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -3735982/126133 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -9960277/337937 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -3890508/131999 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -2441047/82821 during the search for all slope function fixed points. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 38854/1145167. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 38854/1145167. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 38854/1145167. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 38854/1145167. Imminent integer overflow halted evaluation of the slope function at slope -2153606/73069 while computing a fixed point excluded interval about 38854/1145167. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 38854/1145167. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 38854/1145167. Imminent integer overflow halted evaluation of the slope function at slope -6553424/222349 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -10376208/352051 during the search for all slope function fixed points. Imminent integer overflow caused the modular group computation to abort.