These Thurston maps are NET maps for every choice of translation term. They are primitive and have degree 33. 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 0/11, 0/33, 1/33, 1/11, 1/3, 2/3, 1/1, 3/3, 4/3, 5/3, 2/1, 8/3, 3/1, 9/3 4/1, 5/1, 6/1, 7/1, 8/1, 9/1, 10/1, 12/1, 13/1, 14/1, 15/1, 16/1, 18/1, 21/1 27/1, 30/1, 31/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,0.000000) ( 0.000000,0.031746) ( 0.034128,0.034845) ( 0.035340,0.036260) ( 0.037460,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 (-0.011306,0.007666) 0/1 EXTENDED HST ( 0.031722,0.031946) 2/63 HST ( 0.031831,0.031990) 7/219 HST ( 0.031982,0.032058) 4/125 HST ( 0.032007,0.032093) 6/187 HST ( 0.032034,0.032164) 13/405 HST ( 0.032103,0.032116) 7/218 HST ( 0.032152,0.032176) 11/342 HST ( 0.032175,0.032178) 165/5128 HST ( 0.032176,0.032180) 13/404 HST ( 0.032178,0.032200) 14/435 HST ( 0.032186,0.032208) 18/559 HST ( 0.032199,0.032218) 21/652 HST ( 0.032214,0.032226) 27/838 HST ( 0.032225,0.032227) 97/3010 HST ( 0.032226,0.032227) 33/1024 HST ( 0.032222,0.032232) 34/1055 HST ( 0.032228,0.032238) 40/1241 HST ( 0.032234,0.032241) 51/1582 HST ( 0.032239,0.032244) 62/1923 HST ( 0.032242,0.032247) 77/2388 HST ( 0.032245,0.032249) 93/2884 HST ( 0.032247,0.032250) 112/3473 HST ( 0.032249,0.032252) 137/4248 HST ( 0.032251,0.032253) 167/5178 HST ( 0.032253,0.032264) 1/31 EXTENDED HST ( 0.032264,0.032265) 162/5021 HST ( 0.032265,0.032265) 161/4990 HST ( 0.032265,0.032267) 134/4153 HST ( 0.032266,0.032269) 110/3409 HST ( 0.032268,0.032271) 90/2789 HST ( 0.032270,0.032270) 89/2758 HST ( 0.032270,0.032275) 74/2293 HST ( 0.032272,0.032278) 60/1859 HST ( 0.032276,0.032283) 48/1487 HST ( 0.032280,0.032280) 47/1456 HST ( 0.032281,0.032302) 33/1022 HST ( 0.032290,0.032294) 30/929 HST ( 0.032294,0.032305) 24/743 HST ( 0.032303,0.032323) 18/557 HST ( 0.032316,0.032339) 16/495 HST ( 0.032327,0.032349) 12/371 HST ( 0.032348,0.032383) 10/309 HST ( 0.032364,0.032382) 9/278 HST ( 0.032380,0.032386) 25/772 HST ( 0.032384,0.032429) 8/247 HST ( 0.032395,0.032443) 6/185 HST ( 0.032417,0.032474) 11/339 HST ( 0.032456,0.032481) 5/154 HST ( 0.032426,0.032536) 9/277 HST ( 0.032519,0.032522) 4/123 EXTENDED HST ( 0.032532,0.032541) 49/1506 HST ( 0.032537,0.032576) 7/215 HST ( 0.032565,0.032648) 3/92 HST ( 0.032616,0.032699) 5/153 HST ( 0.032690,0.032732) 7/214 HST ( 0.032729,0.032747) 11/336 HST ( 0.032733,0.032771) 15/458 HST ( 0.032769,0.032774) 35/1068 HST ( 0.032772,0.032777) 43/1312 HST ( 0.032775,0.032779) 53/1617 HST ( 0.032777,0.032781) 67/2044 HST ( 0.032780,0.032782) 87/2654 HST ( 0.032781,0.032783) 107/3264 HST ( 0.032782,0.032792) 2/61 EXTENDED HST ( 0.032791,0.032793) 103/3141 HST ( 0.032792,0.032794) 85/2592 HST ( 0.032793,0.032796) 71/2165 HST ( 0.032795,0.032817) 31/945 HST ( 0.032804,0.032813) 21/640 HST ( 0.032813,0.032824) 17/518 HST ( 0.032820,0.032838) 11/335 HST ( 0.032837,0.032856) 9/274 HST ( 0.032847,0.032895) 7/213 HST ( 0.032879,0.032909) 5/152 HST ( 0.032886,0.032937) 8/243 HST ( 0.032931,0.032955) 14/425 HST ( 0.032955,0.032959) 32/971 HST ( 0.032955,0.032964) 44/1335 HST ( 0.032943,0.032987) 119/3610 HST ( 0.032965,0.032969) 3/91 EXTENDED HST ( 0.032984,0.032989) 92/2789 HST ( 0.032987,0.033054) 7/212 HST ( 0.033052,0.033056) 71/2148 HST ( 0.033056,0.033060) 4/121 EXTENDED HST ( 0.033058,0.033083) 25/756 HST ( 0.033070,0.033077) 17/514 HST ( 0.033077,0.033158) 5/151 HST ( 0.033106,0.033179) 7/211 HST ( 0.033166,0.033207) 8/241 HST ( 0.033197,0.033236) 9/271 HST ( 0.033202,0.033263) 12/361 HST ( 0.033217,0.033333) 16/481 HST ( 0.033317,0.033350) 1/30 EXTENDED HST ( 0.033348,0.033359) 56/1679 HST ( 0.033353,0.033362) 46/1379 HST ( 0.033358,0.033381) 30/899 HST ( 0.033371,0.033418) 20/599 HST ( 0.033389,0.033426) 17/509 HST ( 0.033399,0.033435) 12/359 HST ( 0.033428,0.033688) 4/119 HST ( 0.033613,0.033771) 19/564 HST ( 0.033706,0.033710) 3/89 EXTENDED HST ( 0.033740,0.033823) 5/148 HST ( 0.033771,0.033920) 9/266 HST ( 0.033893,0.033904) 2/59 EXTENDED HST ( 0.033906,0.034062) 5/147 HST ( 0.034048,0.034139) 3/88 HST ( 0.034766,0.035838) 2/57 HST ( 0.035859,0.036883) 7/193 HST ( 0.036344,0.041519) 1/27 EXTENDED HST -> HST The supplemental half-space computation shows that these NET maps are rational. SLOPE FUNCTION INFORMATION NUMBER OF FIXED POINTS FOUND: 15 EQUATOR? FIXED POINT c d 0 lambda1 lambda2 lambda1+lambda2 -30/1 1 33 Yes Yes No No 0/1 1 33 Yes Yes No No -960/31 1 33 Yes Yes No No -832/27 1 33 Yes Yes No No -214/7 1 33 Yes Yes No No -640/21 1 33 Yes Yes No No -576/19 1 33 Yes Yes No No -512/17 1 33 Yes Yes No No -896/29 1 33 Yes Yes No No -768/25 1 33 Yes Yes No No -704/23 1 33 Yes Yes No No -448/15 1 33 Yes Yes No No -128/5 1 33 Yes Yes No No -64/3 1 33 Yes Yes No No -256/9 1 33 Yes Yes No No NUMBER OF EQUATORS FOUND: 15 15 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: 6556 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,0.032629) ( 0.032629,0.032708) ( 0.032708,0.032710) ( 0.032710,0.033073) ( 0.033073,0.033073) ( 0.033073,0.033139) ( 0.033139,0.033139) ( 0.033139,0.033408) ( 0.033408,0.033408) ( 0.033408,0.033408) ( 0.033408,0.033408) ( 0.033408,0.033408) ( 0.033408,0.033408) ( 0.033408,0.033408) ( 0.033408,0.033408) ( 0.033408,0.033408) ( 0.033408,0.033409) ( 0.033409,0.033409) ( 0.033410,0.033411) ( 0.033413,0.033413) ( 0.033417,0.033421) ( 0.033426,0.033426) ( 0.033435,0.033435) ( 0.033444,0.033446) ( 0.033457,0.033458) ( 0.033473,0.033473) ( 0.033489,0.033496) ( 0.033519,0.033520) ( 0.033550,0.033564) ( 0.033610,0.033616) ( 0.033708,0.033708) ( 0.033898,0.033899) ( 0.034423,0.034519) ( 0.035639,0.035794) ( 0.037033,0.037042) ( 0.038331,0.038601) ( 0.039991,0.040009) ( 0.041363,0.041995) ( 0.043421,0.043536) ( 0.045093,0.045845) ( 0.047550,0.047688) ( 0.049062,0.051037) ( 0.052573,0.052692) ( 0.054662,0.056537) ( 0.058719,0.058929) ( 0.060932,0.064257) ( 0.066569,0.066768) ( 0.069246,0.073903) ( 0.076774,0.077068) ( 0.079070,0.088398) ( 0.090659,0.091160) ( 0.094708,0.106312) ( 0.110807,0.111417) ( 0.116041,0.136170) ( 0.142345,0.143383) ( 0.150442,0.188235) ( 0.198795,0.201220) ( 0.215190,0.301887) ( 0.330544,0.336261) ( 0.390805,0.711111) ( 0.976600,1.025347) (34.750000,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,b^-1>(2,33)(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)", "b=(1,33)(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,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>(2,33)(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)", "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,c>(1,2)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=(1,33)(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=(1,33)(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,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>(2,33)(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 lambda2: NewSphereMachine( "a=(1,33)(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,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>(2,33)(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)", "c=(1,33)(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=(1,32)(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)", "a*b*c*d"); When the translation term of the affine map is lambda1+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=(1,33)(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)(32,33)", "a*b*c*d"); ****************************INTEGER OVERFLOW REPORT***************************** Imminent integer overflow halted evaluation of the slope function at slope -2111326/68891 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -1156252/37819 while computing a fixed point excluded interval about 16935/517759. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 27751/848407. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 27751/848407. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 27751/848407. Imminent integer overflow halted evaluation of the slope function at slope -2120426/69361 while computing a fixed point excluded interval about 67576/2065857. Imminent integer overflow halted evaluation of the slope function at slope -9955184/326153 while computing a fixed point excluded interval about 109636/3346425. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 42538/1294069. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 42538/1294069. Imminent integer overflow halted evaluation of the slope function at slope -2002996/65843 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -1320930/43687 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -2018782/66767 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -1075088/35627 while computing a fixed point excluded interval about 34248/1033475. Imminent integer overflow halted evaluation of the slope function at slope -1896602/62851 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -2142598/71003 while computing a fixed point excluded interval about 62437/1884109. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 62437/1884109. Imminent integer overflow halted evaluation of the slope function at slope -2659576/88135 while computing a fixed point excluded interval about 62437/1884109. Imminent integer overflow halted evaluation of the slope function at slope -1658234/55297 while computing a fixed point excluded interval about 54888/1645969. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 2474/74055. Imminent integer overflow halted evaluation of the slope function at slope -2726824/91097 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -1511716/50503 during the search for all slope function fixed points. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 12738/381289. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 12738/381289. Imminent integer overflow halted evaluation of the slope function at slope -3403374/113699 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -7329172/244851 while computing a fixed point excluded interval about 77371/2315961. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 77371/2315961. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 77371/2315961. Imminent integer overflow halted evaluation of the slope function at slope -13104961/437807 while computing a fixed point excluded interval about 77371/2315961. Imminent integer overflow halted evaluation of the slope function at slope -18118172/605287 while computing a fixed point excluded interval about 77371/2315961. Imminent integer overflow halted evaluation of the slope function at slope -4984984/166537 while computing a fixed point excluded interval about 77371/2315961. Imminent integer overflow halted evaluation of the slope function at slope -5366273/179275 while computing a fixed point excluded interval about 77371/2315961. Imminent integer overflow halted evaluation of the slope function at slope -23512672/785505 while computing a fixed point excluded interval about 77371/2315961. Imminent integer overflow halted evaluation of the slope function at slope -5747562/192013 while computing a fixed point excluded interval about 77371/2315961. Imminent integer overflow caused the modular group computation to abort.