These Thurston maps are NET maps for every choice of translation term. They are primitive and have degree 27. 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 0/9, 0/27, 1/27, 1/9, 1/3, 2/3, 1/1, 3/3, 4/3, 2/1, 6/3, 7/3, 3/1, 4/1, 5/1 6/1, 7/1, 8/1, 10/1, 11/1, 12/1, 13/1, 15/1, 21/1, 24/1, 25/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,0.000000) ( 0.000000,0.039216) ( 0.042857,0.044118) ( 0.044159,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 (-0.015625,0.010204) 0/1 EXTENDED HST ( 0.039157,0.039535) 2/51 HST ( 0.039332,0.039599) 7/177 HST ( 0.039558,0.039708) 4/101 HST ( 0.039607,0.039753) 6/151 HST ( 0.039722,0.039776) 13/327 HST ( 0.039761,0.039783) 7/176 HST ( 0.039776,0.039794) 22/553 HST ( 0.039794,0.039854) 8/201 HST ( 0.039818,0.039889) 11/276 HST ( 0.039875,0.039913) 15/376 HST ( 0.039911,0.039916) 92/2305 HST ( 0.039913,0.039918) 19/476 HST ( 0.039910,0.039938) 20/501 HST ( 0.039935,0.039945) 26/651 HST ( 0.039921,0.039962) 29/726 HST ( 0.039953,0.039969) 42/1051 HST ( 0.039964,0.039976) 52/1301 HST ( 0.039970,0.039980) 66/1651 HST ( 0.039979,0.039982) 82/2051 HST ( 0.039978,0.039985) 87/2176 HST ( 0.039985,0.039985) 109/2726 HST ( 0.039983,0.039988) 110/2751 HST ( 0.039986,0.039991) 138/3451 HST ( 0.039989,0.040011) 1/25 EXTENDED HST ( 0.040011,0.040017) 114/2849 HST ( 0.040014,0.040021) 92/2299 HST ( 0.040017,0.040025) 75/1874 HST ( 0.040021,0.040031) 61/1524 HST ( 0.040026,0.040039) 49/1224 HST ( 0.040033,0.040049) 39/974 HST ( 0.040041,0.040085) 27/674 HST ( 0.040061,0.040069) 24/599 HST ( 0.040069,0.040071) 23/574 HST ( 0.040070,0.040098) 18/449 HST ( 0.040092,0.040120) 16/399 HST ( 0.040104,0.040141) 12/299 HST ( 0.040137,0.040145) 148/3687 HST ( 0.040141,0.040218) 10/249 HST ( 0.040171,0.040244) 8/199 HST ( 0.040202,0.040253) 7/174 HST ( 0.040245,0.040263) 19/472 HST ( 0.040260,0.040275) 241/5985 HST ( 0.040267,0.040270) 6/149 EXTENDED HST ( 0.040243,0.040321) 17/422 HST ( 0.040287,0.040322) 16/397 HST ( 0.040302,0.040346) 5/124 HST ( 0.040276,0.040430) 14/347 HST ( 0.040356,0.040529) 4/99 HST ( 0.040466,0.040607) 3/74 HST ( 0.040580,0.040678) 5/123 HST ( 0.040652,0.040732) 7/172 HST ( 0.040731,0.040752) 11/270 HST ( 0.040749,0.040762) 13/319 HST ( 0.040753,0.040786) 17/417 HST ( 0.040783,0.040795) 29/711 HST ( 0.040793,0.040801) 41/1005 HST ( 0.040797,0.040805) 55/1348 HST ( 0.040803,0.040807) 73/1789 HST ( 0.040806,0.040827) 2/49 EXTENDED HST ( 0.040825,0.040829) 77/1886 HST ( 0.040827,0.040833) 61/1494 HST ( 0.040830,0.040837) 51/1249 HST ( 0.040833,0.040840) 41/1004 HST ( 0.040837,0.040846) 33/808 HST ( 0.040843,0.040857) 27/661 HST ( 0.040848,0.040865) 19/465 HST ( 0.040861,0.040873) 17/416 HST ( 0.040869,0.040892) 13/318 HST ( 0.040891,0.040903) 11/269 HST ( 0.040893,0.040925) 9/220 HST ( 0.040886,0.040968) 23/562 HST ( 0.040929,0.041031) 5/122 HST ( 0.040984,0.041096) 11/268 HST ( 0.041091,0.041101) 3/73 EXTENDED HST ( 0.041097,0.041109) 70/1703 HST ( 0.041104,0.041131) 25/608 HST ( 0.041120,0.041131) 22/535 HST ( 0.041121,0.041237) 7/170 HST ( 0.041226,0.041477) 4/97 HST ( 0.041454,0.041508) 37/892 HST ( 0.041480,0.041547) 10/241 HST ( 0.041503,0.041567) 15/361 HST ( 0.041525,0.041601) 18/433 HST ( 0.041574,0.041635) 27/649 HST ( 0.041624,0.041645) 54/1297 HST ( 0.041636,0.041699) 1/24 EXTENDED HST ( 0.041667,0.041935) 12/287 HST ( 0.041818,0.042035) 6/143 HST ( 0.042006,0.042286) 3/71 HST ( 0.042276,0.042329) 14/331 HST ( 0.042296,0.042440) 5/118 HST ( 0.042313,0.042587) 9/212 HST ( 0.042542,0.042564) 2/47 EXTENDED HST ( 0.042553,0.042857) 5/117 HST ( 0.042783,0.042941) 3/70 HST ( 0.043962,0.044296) 3/68 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 -24/1 1 27 Yes Yes No No 0/1 1 27 Yes Yes No No -122/5 1 27 Yes Yes No No -624/25 1 27 Yes Yes No No -416/17 1 27 Yes Yes No No -260/11 1 27 Yes Yes No No -572/23 1 27 Yes Yes No No -520/21 1 27 Yes Yes No No -468/19 1 27 Yes Yes No No -364/15 1 27 Yes Yes No No -208/9 1 27 Yes Yes No No -156/7 1 27 Yes Yes No No -104/5 1 27 Yes Yes No No -52/3 1 27 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: 5811 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,0.041533) ( 0.041533,0.041533) ( 0.041533,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^-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>(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)", "b=(1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,15)", "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*d>(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)", "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,c,c,c,c,c,c,c,c,c,c,c,c>(1,2)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=(1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,15)", "b=(1,26)(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=(1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,15)", "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,c,c,c,c,c,c,c,c,c,c,c,c*d>(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,15)", "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,c,c,c,c,c,c,c,c,c,c,c,c*d>(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)", "c=(1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,15)", "d=(1,26)(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 lambda1+lambda2: NewSphereMachine( "a=(1,26)(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=(1,27)(2,26)(3,25)(4,24)(5,23)(6,22)(7,21)(8,20)(9,19)(10,18)(11,17)(12,16)(13,15)", "c=(1,26)(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=(1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(26,27)", "a*b*c*d"); ****************************INTEGER OVERFLOW REPORT***************************** Imminent integer overflow halted evaluation of the slope function at slope -1767240/72433 while computing a fixed point excluded interval about 37605/917497. Imminent integer overflow halted evaluation of the slope function at slope -15277750/634533 while computing a fixed point excluded interval about 311486/7499697. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 114949/2767645. Imminent integer overflow halted evaluation of the slope function at slope -5813646/241459 while computing a fixed point excluded interval about 114949/2767645. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 138071/3324357. Imminent integer overflow halted evaluation of the slope function at slope -64538648/2680493 while computing a fixed point excluded interval about 1472537/35454503. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 161193/3881069. Imminent integer overflow halted evaluation of the slope function at slope -8597206/357069 while computing a fixed point excluded interval about 161193/3881069. Imminent integer overflow halted evaluation of the slope function at slope -9153918/380191 while computing a fixed point excluded interval about 161193/3881069. Imminent integer overflow halted evaluation of the slope function at slope -21894634/909353 while computing a fixed point excluded interval about 161193/3881069. Imminent integer overflow halted evaluation of the slope function at slope -10267342/426435 while computing a fixed point excluded interval about 161193/3881069. Imminent integer overflow halted evaluation of the slope function at slope -14784698/614055 while computing a fixed point excluded interval about 312808/7531527. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 23783/572627. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 23783/572627. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 35344/850983. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 35344/850983. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 58466/1407695. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 58466/1407695. Imminent integer overflow halted evaluation of the slope function at slope -3252896/135103 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -36574572/1519057 while computing a fixed point excluded interval about 650930/15672543. Imminent integer overflow halted evaluation of the slope function at slope -2792200/115971 during the search for all slope function fixed points. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 32291/777461. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 72834/1753603. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 113377/2729745. Imminent integer overflow halted evaluation of the slope function at slope -4252538/179219 while computing a fixed point excluded interval about 162102/3846383. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 77536/1839787.