These Thurston maps are NET maps for every choice of translation term. They are primitive and have degree 38. 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 0/19, 0/38, 1/38, 1/19, 1/2, 1/1, 2/2, 3/2, 4/2, 5/2, 3/1, 6/2, 7/2, 8/2 9/2, 5/1, 10/2, 13/2, 7/1, 16/2, 9/1, 11/1, 13/1, 15/1, 17/1, 23/1, 29/1 35/1, 36/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,0.000000) ( 0.000000,0.027397) ( 0.029172,0.029751) ( 0.030133,0.030579) ( 0.030980,0.031633) ( 0.032595,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 (-0.006773,0.005446) 0/1 EXTENDED HST ( 0.027361,0.027525) 2/73 HST ( 0.027518,0.027598) 7/254 HST ( 0.027583,0.027632) 4/145 HST ( 0.027629,0.027637) 37/1339 HST ( 0.027634,0.027634) 16/579 EXTENDED HST ( 0.027636,0.027640) 11/398 HST ( 0.027632,0.027648) 28/1013 HST ( 0.027642,0.027677) 6/217 HST ( 0.027672,0.027703) 8/289 HST ( 0.027694,0.027712) 11/397 HST ( 0.027705,0.027727) 12/433 HST ( 0.027724,0.027738) 16/577 HST ( 0.027733,0.027744) 20/721 HST ( 0.027742,0.027746) 137/4938 HST ( 0.027744,0.027744) 23/829 EXTENDED HST ( 0.027745,0.027747) 414/14921 HST ( 0.027746,0.027746) 195/7028 HST ( 0.027746,0.027746) 122/4397 HST ( 0.027746,0.027746) 49/1766 HST ( 0.027747,0.027747) 25/901 HST ( 0.027742,0.027753) 26/937 HST ( 0.027748,0.027757) 31/1117 HST ( 0.027757,0.027757) 37/1333 HST ( 0.027738,0.027778) 38/1369 HST ( 0.027775,0.027781) 1/36 EXTENDED HST ( 0.027778,0.027817) 36/1295 HST ( 0.027800,0.027800) 35/1259 HST ( 0.027800,0.027809) 29/1043 HST ( 0.027805,0.027807) 28/1007 HST ( 0.027805,0.027816) 24/863 HST ( 0.027811,0.027812) 23/827 HST ( 0.027813,0.027813) 22/791 EXTENDED HST ( 0.027815,0.027815) 21/755 EXTENDED HST ( 0.027815,0.027829) 17/611 HST ( 0.027826,0.027826) 16/575 EXTENDED HST ( 0.027829,0.027830) 15/539 HST ( 0.027824,0.027838) 29/1042 HST ( 0.027831,0.027847) 13/467 HST ( 0.027840,0.027863) 10/359 HST ( 0.027857,0.027885) 8/287 HST ( 0.027885,0.027892) 7/251 HST ( 0.027891,0.027949) 5/179 HST ( 0.027937,0.027962) 9/322 HST ( 0.027955,0.027973) 21/751 HST ( 0.027971,0.027973) 4/143 EXTENDED HST ( 0.027962,0.028015) 11/393 HST ( 0.027995,0.028005) 7/250 HST ( 0.028004,0.028028) 13/464 HST ( 0.028019,0.028059) 3/107 HST ( 0.028057,0.028066) 11/392 HST ( 0.028062,0.028070) 46/1639 HST ( 0.028067,0.028067) 27/962 EXTENDED HST ( 0.028070,0.028070) 8/285 EXTENDED HST ( 0.028058,0.028085) 77/2743 HST ( 0.028072,0.028074) 45/1603 HST ( 0.028073,0.028077) 21/748 HST ( 0.028077,0.028083) 13/463 HST ( 0.028081,0.028091) 33/1175 HST ( 0.028086,0.028110) 5/178 HST ( 0.028110,0.028111) 54/1921 HST ( 0.028111,0.028136) 7/249 HST ( 0.028120,0.028160) 13/462 HST ( 0.028157,0.028162) 43/1527 HST ( 0.028161,0.028163) 57/2024 HST ( 0.028162,0.028164) 69/2450 HST ( 0.028163,0.028165) 83/2947 HST ( 0.028165,0.028166) 109/3870 HST ( 0.028165,0.028167) 131/4651 HST ( 0.028166,0.028167) 175/6213 HST ( 0.028167,0.028167) 207/7349 HST ( 0.028167,0.028168) 251/8911 HST ( 0.028168,0.028168) 327/11609 HST ( 0.028168,0.028170) 2/71 EXTENDED HST ( 0.028170,0.028170) 319/11324 HST ( 0.028170,0.028171) 267/9478 HST ( 0.028171,0.028171) 223/7916 HST ( 0.028171,0.028172) 173/6141 HST ( 0.028171,0.028173) 133/4721 HST ( 0.028172,0.028173) 111/3940 HST ( 0.028173,0.028175) 85/3017 HST ( 0.028174,0.028175) 73/2591 HST ( 0.028175,0.028177) 59/2094 HST ( 0.028176,0.028179) 49/1739 HST ( 0.028177,0.028181) 39/1384 HST ( 0.028179,0.028185) 33/1171 HST ( 0.028181,0.028191) 25/887 HST ( 0.028186,0.028190) 23/816 HST ( 0.028188,0.028192) 19/674 HST ( 0.028191,0.028200) 17/603 HST ( 0.028194,0.028205) 13/461 HST ( 0.028201,0.028209) 11/390 HST ( 0.028204,0.028217) 29/1028 HST ( 0.028213,0.028213) 9/319 EXTENDED HST ( 0.028205,0.028232) 25/886 HST ( 0.028218,0.028223) 16/567 HST ( 0.028220,0.028231) 7/248 HST ( 0.028228,0.028272) 5/177 HST ( 0.028264,0.028283) 19/672 HST ( 0.028277,0.028284) 11/389 HST ( 0.028274,0.028301) 17/601 HST ( 0.028300,0.028304) 3/106 EXTENDED HST ( 0.028295,0.028357) 10/353 HST ( 0.028337,0.028344) 7/247 HST ( 0.028349,0.028367) 19/670 HST ( 0.028366,0.028368) 103/3631 HST ( 0.028368,0.028370) 4/141 EXTENDED HST ( 0.028361,0.028405) 13/458 HST ( 0.028389,0.028431) 5/176 HST ( 0.028427,0.028480) 6/211 HST ( 0.028475,0.028488) 9/316 HST ( 0.028470,0.028504) 10/351 HST ( 0.028500,0.028518) 12/421 HST ( 0.028517,0.028530) 16/561 HST ( 0.028525,0.028540) 20/701 HST ( 0.028537,0.028545) 27/946 HST ( 0.028543,0.028551) 32/1121 HST ( 0.028551,0.028594) 1/35 EXTENDED HST ( 0.028592,0.028600) 33/1154 HST ( 0.028597,0.028606) 27/944 HST ( 0.028602,0.028631) 20/699 HST ( 0.028614,0.028625) 18/629 HST ( 0.028619,0.028633) 15/524 HST ( 0.028629,0.028665) 12/419 HST ( 0.028645,0.028691) 9/314 HST ( 0.028672,0.028705) 7/244 HST ( 0.028692,0.028718) 43/1498 HST ( 0.028708,0.028709) 6/209 EXTENDED HST ( 0.028713,0.028756) 5/174 HST ( 0.028732,0.028789) 13/452 HST ( 0.028773,0.028843) 4/139 HST ( 0.028789,0.028910) 3/104 HST ( 0.028899,0.028954) 7/242 HST ( 0.028948,0.028959) 15/518 HST ( 0.028953,0.028963) 32/1105 HST ( 0.028960,0.028967) 17/587 HST ( 0.028929,0.029000) 23/794 HST ( 0.028982,0.028989) 2/69 EXTENDED HST ( 0.028963,0.029088) 11/379 HST ( 0.029026,0.029244) 3/103 HST ( 0.029714,0.029807) 5/168 HST ( 0.029491,0.030137) 11/369 HST ( 0.029846,0.029856) 2/67 EXTENDED HST ( 0.030014,0.031483) 3/98 HST ( 0.030765,0.030774) 2/65 EXTENDED HST ( 0.031570,0.032308) 2/63 HST ( 0.032214,0.032302) 1/31 EXTENDED HST ( 0.031325,0.036131) 4/123 HST The supplemental half-space computation shows that these NET maps are rational. SLOPE FUNCTION INFORMATION NUMBER OF FIXED POINTS FOUND: 9 EQUATOR? FIXED POINT c d 0 lambda1 lambda2 lambda1+lambda2 -36/1 1 38 Yes Yes No No 0/1 1 38 Yes Yes No No -106/3 1 38 Yes Yes No No -178/5 1 38 Yes Yes No No -962/27 1 38 Yes Yes No No -816/23 1 38 Yes Yes No No -518/15 1 38 Yes Yes No No -74/3 1 38 Yes Yes No No -222/7 1 38 Yes Yes No No NUMBER OF EQUATORS FOUND: 9 9 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: 2082 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,0.027777) ( 0.027777,0.027778) ( 0.028570,0.028573) ( 0.029367,0.029459) ( 0.030280,0.030328) ( 0.031200,0.031303) ( 0.032252,0.032264) ( 0.033220,0.033454) ( 0.034211,0.034774) ( 0.035455,0.035992) ( 0.037015,0.037060) ( 0.038160,0.038784) ( 0.039796,0.040217) ( 0.040710,0.042725) ( 0.043401,0.043560) ( 0.044725,0.046250) ( 0.046875,0.048429) ( 0.048995,0.051105) ( 0.052525,0.052740) ( 0.054242,0.057011) ( 0.057353,0.060458) ( 0.060937,0.064236) ( 0.066468,0.066856) ( 0.068182,0.087678) ( 0.090570,0.091269) ( 0.095122,0.136029) ( 0.141978,0.143778) ( 0.151163,0.303279) ( 0.328533,0.338554) ( 0.390000,1.850000) (39.750000,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,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,b^-1,b^-1>(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)", "b=(1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)", "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^-1,c^-1,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c*d>(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)", "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^-1,c^-1,c^-1,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c>(1,2)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=(1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)", "b=(1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)", "c=(1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)", "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^-1,c^-1,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c*d>(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)", "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^-1,c^-1,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c*d>(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)", "c=(1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)", "d=(1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)", "b=(1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)", "c=(1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)", "d=(1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(37,38)", "a*b*c*d"); ****************************INTEGER OVERFLOW REPORT***************************** Imminent integer overflow halted evaluation of the slope function at slope -1225688/34011 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -1298700/36037 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -1376020/38189 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -3794796/105373 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -1264956/35125 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -1259410/34971 during the search for all slope function fixed points. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 719/25885. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 719/25885. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 719/25885. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 719/25885. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 719/25885. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 719/25885. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 2461/88599. Imminent integer overflow halted evaluation of the slope function at slope -6801958/188937 during the search for all slope function fixed points. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 20926/753361. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 20926/753361. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 20926/753361. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 20926/753361. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 20926/753361. Imminent integer overflow halted evaluation of the slope function at slope -3857204/107141 while computing a fixed point excluded interval about 20926/753361. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 20926/753361. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 20926/753361. Imminent integer overflow halted evaluation of the slope function at slope -5424192/150667 while computing a fixed point excluded interval about 20926/753361. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 20926/753361. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 20926/753361. Imminent integer overflow halted evaluation of the slope function at slope -2471014/68637 while computing a fixed point excluded interval about 20926/753361. Imminent integer overflow halted evaluation of the slope function at slope -6801958/188937 during the search for all slope function fixed points. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 847/30493. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 847/30493. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 847/30493. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 847/30493. Imminent integer overflow caused the modular group computation to abort.