These Thurston maps are NET maps for every choice of translation term. They are primitive and have degree 25. 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 0/25, 1/25, 1/5, 2/5, 3/5, 1/1, 2/1, 3/1, 4/1, 6/1, 7/1, 8/1, 9/1, 11/1 12/1, 13/1, 16/1, 19/1, 22/1, 23/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,0.000000) ( 0.000000,0.042553) ( 0.046253,0.047017) ( 0.049180,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 (-0.015371,0.010746) 0/1 EXTENDED HST ( 0.042536,0.042570) 2/47 EXTENDED HST ( 0.042234,0.042885) 17/399 HST ( 0.042607,0.042755) 7/164 HST ( 0.042683,0.043011) 3/70 HST ( 0.042827,0.043063) 4/93 HST ( 0.043011,0.043189) 5/116 HST ( 0.043147,0.043269) 7/162 HST ( 0.043253,0.043290) 9/208 HST ( 0.043250,0.043339) 10/231 HST ( 0.043330,0.043376) 14/323 HST ( 0.043360,0.043398) 19/438 HST ( 0.043385,0.043408) 24/553 HST ( 0.043377,0.043430) 27/622 HST ( 0.043420,0.043440) 40/921 HST ( 0.043432,0.043448) 49/1128 HST ( 0.043442,0.043454) 62/1427 HST ( 0.043449,0.043460) 79/1818 HST ( 0.043459,0.043460) 101/2324 HST ( 0.043456,0.043464) 102/2347 HST ( 0.043461,0.043495) 1/23 EXTENDED HST ( 0.043493,0.043501) 102/2345 HST ( 0.043497,0.043506) 81/1862 HST ( 0.043502,0.043513) 66/1517 HST ( 0.043507,0.043521) 53/1218 HST ( 0.043514,0.043535) 40/919 HST ( 0.043526,0.043527) 39/896 HST ( 0.043527,0.043552) 30/689 HST ( 0.043543,0.043545) 29/666 HST ( 0.043544,0.043576) 24/551 HST ( 0.043557,0.043578) 21/482 HST ( 0.043571,0.043608) 18/413 HST ( 0.043584,0.043624) 15/344 HST ( 0.043608,0.043620) 14/321 HST ( 0.043619,0.043682) 11/252 HST ( 0.043659,0.043808) 8/183 HST ( 0.043742,0.043793) 7/160 HST ( 0.043794,0.043798) 6/137 EXTENDED HST ( 0.043796,0.043919) 5/114 HST ( 0.043890,0.044047) 4/91 HST ( 0.044025,0.044199) 3/68 HST ( 0.044177,0.044260) 5/113 HST ( 0.044253,0.044361) 7/158 HST ( 0.044349,0.044380) 13/293 HST ( 0.044358,0.044401) 17/383 HST ( 0.044391,0.044412) 23/518 HST ( 0.044406,0.044419) 31/698 HST ( 0.044415,0.044424) 39/878 HST ( 0.044420,0.044430) 49/1103 HST ( 0.044428,0.044433) 69/1553 HST ( 0.044431,0.044436) 89/2003 HST ( 0.044434,0.044437) 115/2588 HST ( 0.044436,0.044439) 141/3173 HST ( 0.044438,0.044451) 2/45 EXTENDED HST ( 0.044450,0.044453) 137/3082 HST ( 0.044452,0.044455) 111/2497 HST ( 0.044453,0.044459) 87/1957 HST ( 0.044456,0.044462) 71/1597 HST ( 0.044458,0.044465) 57/1282 HST ( 0.044462,0.044471) 47/1057 HST ( 0.044466,0.044476) 37/832 HST ( 0.044472,0.044485) 31/697 HST ( 0.044476,0.044492) 25/562 HST ( 0.044484,0.044639) 7/157 HST ( 0.044637,0.044641) 102/2285 HST ( 0.044640,0.044645) 5/112 EXTENDED HST ( 0.044639,0.044674) 28/627 HST ( 0.044657,0.044776) 8/179 HST ( 0.044700,0.044919) 3/67 HST ( 0.044893,0.044951) 23/512 HST ( 0.044939,0.044948) 4/89 EXTENDED HST ( 0.044945,0.044970) 37/823 HST ( 0.044959,0.044991) 13/289 HST ( 0.044990,0.045010) 9/200 HST ( 0.044906,0.045089) 14/311 HST ( 0.045017,0.045153) 5/111 HST ( 0.045131,0.045232) 7/155 HST ( 0.045220,0.045299) 10/221 HST ( 0.045181,0.045455) 14/309 HST ( 0.045415,0.045496) 1/22 EXTENDED HST ( 0.045495,0.045514) 41/901 HST ( 0.045505,0.045532) 33/725 HST ( 0.045518,0.045641) 16/351 HST ( 0.045589,0.045698) 13/285 HST ( 0.045618,0.045886) 6/131 HST ( 0.045833,0.045996) 5/109 HST ( 0.045918,0.046196) 3/65 HST ( 0.046122,0.046274) 17/368 HST ( 0.046763,0.047619) 4/85 HST ( 0.047563,0.047678) 1/21 EXTENDED HST ( 0.047495,0.048168) 14/293 HST ( 0.047782,0.047872) 12/251 HST ( 0.047817,0.047930) 9/188 HST ( 0.047872,0.048023) 7/146 HST ( 0.047970,0.048193) 6/125 HST ( 0.048011,0.048309) 4/83 HST ( 0.048286,0.048524) 3/62 HST ( 0.048484,0.050675) 3/61 HST The supplemental half-space computation shows that these NET maps are rational. SLOPE FUNCTION INFORMATION NUMBER OF FIXED POINTS FOUND: 14 EQUATOR? FIXED POINT c d 0 lambda1 lambda2 lambda1+lambda2 -22/1 1 25 Yes Yes No No -16/1 1 25 Yes Yes No No 0/1 1 25 Yes Yes No No -64/3 1 25 Yes Yes No No -112/5 1 25 Yes Yes No No -160/7 1 25 Yes Yes No No -384/17 1 25 Yes Yes No No -338/15 1 25 Yes Yes No No -528/23 1 25 Yes Yes No No -432/19 1 25 Yes Yes No No -288/13 1 25 Yes Yes No No -240/11 1 25 Yes Yes No No -144/7 1 25 Yes Yes No No -96/5 1 25 Yes Yes No No NUMBER OF EQUATORS FOUND: 14 14 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: 5099 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,0.044379) ( 0.044379,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^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1>(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)", "b=(1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)", "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*d>(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)", "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,c,c,c,c,c,c,c,c,c,c,c>(1,2)(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: NewSphereMachine( "a=(1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)", "b=(1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)", "c=(1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)", "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,c,c,c,c,c,c,c,c,c,c,c*d>(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)", "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,c,c,c,c,c,c,c,c,c,c,c*d>(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)", "c=(1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)", "d=(1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)", "b=(1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)", "c=(1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)", "d=(1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(24,25)", "a*b*c*d"); ****************************INTEGER OVERFLOW REPORT***************************** Imminent integer overflow halted evaluation of the slope function at slope -4325398/191957 while computing a fixed point excluded interval about 190796/4299237. Imminent integer overflow halted evaluation of the slope function at slope -4399428/200807 while computing a fixed point excluded interval about 190496/4173527. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 118571/2597741. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 118571/2597741. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 42789/937453. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 42789/937453. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 80680/1767597. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 80680/1767597. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 107745/2360557. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 107745/2360557.