These Thurston maps are NET maps for every choice of translation term. They are primitive and have degree 30. 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 0/10, 0/15, 0/30, 1/30, 1/15, 1/10, 1/6, 1/5, 1/3, 2/6, 1/2, 3/5, 4/5, 1/1 2/2, 3/3, 4/3, 3/2, 4/2, 7/3, 8/3, 3/1, 6/2, 7/2, 9/2, 12/2, 7/1, 9/1, 11/1 13/1, 27/1, 28/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,0.000000) ( 0.000000,0.035088) ( 0.038109,0.039158) ( 0.039674,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 (-0.013181,0.008782) 0/1 EXTENDED HST ( 0.035008,0.035311) 2/57 HST ( 0.035254,0.035442) 4/113 HST ( 0.035430,0.035464) 14/395 HST ( 0.035447,0.035473) 5/141 HST ( 0.035455,0.035514) 16/451 HST ( 0.035481,0.035521) 6/169 HST ( 0.035453,0.035614) 7/197 HST ( 0.035613,0.035620) 13/365 HST ( 0.035616,0.035638) 14/393 HST ( 0.035630,0.035647) 50/1403 HST ( 0.035638,0.035641) 17/477 HST ( 0.035613,0.035679) 19/533 HST ( 0.035678,0.035679) 36/1009 HST ( 0.035679,0.035680) 37/1037 HST ( 0.035680,0.035682) 38/1065 HST ( 0.035676,0.035688) 40/1121 HST ( 0.035683,0.035693) 97/2718 HST ( 0.035688,0.035689) 49/1373 HST ( 0.035691,0.035697) 62/1737 HST ( 0.035697,0.035698) 75/2101 HST ( 0.035691,0.035705) 79/2213 HST ( 0.035705,0.035705) 143/4005 HST ( 0.035705,0.035706) 145/4061 HST ( 0.035706,0.035706) 146/4089 HST ( 0.035704,0.035707) 147/4117 HST ( 0.035706,0.035723) 1/28 EXTENDED HST ( 0.035714,0.035797) 29/811 HST ( 0.035758,0.035774) 25/699 HST ( 0.035766,0.035786) 20/559 HST ( 0.035780,0.035784) 19/531 HST ( 0.035783,0.035914) 8/223 HST ( 0.035887,0.036030) 5/139 HST ( 0.035995,0.036091) 4/111 HST ( 0.036074,0.036142) 10/277 HST ( 0.036110,0.036186) 3/83 HST ( 0.036137,0.036244) 30/829 HST ( 0.036190,0.036196) 19/525 HST ( 0.036199,0.036200) 8/221 EXTENDED HST ( 0.036213,0.036252) 5/138 HST ( 0.036232,0.036290) 7/193 HST ( 0.036271,0.036307) 9/248 HST ( 0.036304,0.036322) 13/358 HST ( 0.036313,0.036339) 17/468 HST ( 0.036324,0.036354) 27/743 HST ( 0.036350,0.036377) 2/55 EXTENDED HST ( 0.036375,0.036382) 45/1237 HST ( 0.036379,0.036386) 35/962 HST ( 0.036383,0.036385) 33/907 HST ( 0.036385,0.036408) 19/522 HST ( 0.036401,0.036414) 15/412 HST ( 0.036412,0.036436) 11/302 HST ( 0.036432,0.036493) 7/192 HST ( 0.036480,0.036508) 5/137 HST ( 0.036504,0.036529) 18/493 HST ( 0.036515,0.036639) 3/82 HST ( 0.036602,0.036687) 7/191 HST ( 0.036682,0.036693) 35/954 HST ( 0.036685,0.036701) 79/2153 HST ( 0.036696,0.036698) 4/109 EXTENDED HST ( 0.036649,0.036827) 17/463 HST ( 0.036723,0.036749) 9/245 HST ( 0.036746,0.036781) 5/136 HST ( 0.036823,0.036864) 7/190 HST ( 0.036819,0.036897) 8/217 HST ( 0.036854,0.036915) 10/271 HST ( 0.036910,0.036947) 12/325 HST ( 0.036940,0.036967) 16/433 HST ( 0.036961,0.036983) 20/541 HST ( 0.036977,0.036995) 26/703 HST ( 0.036993,0.037084) 1/27 EXTENDED HST ( 0.037076,0.037093) 14847/400357 HST ( 0.037085,0.037109) 24/647 HST ( 0.037095,0.037105) 22/593 HST ( 0.037100,0.037170) 14/377 HST ( 0.037140,0.037146) 13/350 HST ( 0.037146,0.037158) 12/323 HST ( 0.037160,0.037222) 9/242 HST ( 0.037192,0.037306) 7/188 HST ( 0.037255,0.037377) 5/134 HST ( 0.037375,0.037391) 4/107 EXTENDED HST ( 0.037381,0.037404) 39/1043 HST ( 0.037393,0.037622) 3/80 HST ( 0.037577,0.038027) 2/53 HST ( 0.037878,0.038202) 7/184 HST ( 0.038092,0.038098) 4/105 EXTENDED HST ( 0.039101,0.039231) 15/383 HST ( 0.039204,0.039227) 2/51 EXTENDED HST ( 0.039188,0.039285) 37/943 HST ( 0.039237,0.039745) 3/76 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 -28/1 1 30 Yes Yes No No 0/1 1 30 Yes Yes No No -522/19 1 30 Yes Yes No No -406/15 1 30 Yes Yes No No -754/27 1 30 Yes Yes No No -696/25 1 30 Yes Yes No No -638/23 1 30 Yes Yes No No -580/21 1 30 Yes Yes No No -464/17 1 30 Yes Yes No No -348/13 1 30 Yes Yes No No -290/11 1 30 Yes Yes No No -232/9 1 30 Yes Yes No No -174/7 1 30 Yes Yes No No -116/5 1 30 Yes Yes No No -58/3 1 30 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: 8540 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,0.036855) ( 0.036855,0.037250) ( 0.037250,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^-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>(2,30)(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=(1,30)(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)", "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*d>(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=<1,1,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>(1,2)(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: NewSphereMachine( "a=(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)", "b=(1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)", "c=(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)", "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,c,c,c,c,c,c,c,c,c,c,c,c,c,c*d>(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 lambda2: NewSphereMachine( "a=(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)", "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,c,c,c,c,c,c,c,c,c,c,c,c,c,c*d>(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=(1,30)(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)", "d=(1,29)(2,28)(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+lambda2: NewSphereMachine( "a=(1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)", "b=(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)", "c=(1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)", "d=(1,28)(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)(29,30)", "a*b*c*d"); ****************************INTEGER OVERFLOW REPORT***************************** Imminent integer overflow halted evaluation of the slope function at slope -1436372/51269 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -7631892/272557 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -3622726/131357 while computing a fixed point excluded interval about 96437/2659659. Imminent integer overflow halted evaluation of the slope function at slope -6264883/227167 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -2081866/75947 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -2230604/81373 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -2594356/95375 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -2099434/78203 while computing a fixed point excluded interval about 35859/962669. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 55314/1484957. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 55314/1484957. Imminent integer overflow halted evaluation of the slope function at slope -3144010/117113 while computing a fixed point excluded interval about 55314/1484957. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 55314/1484957. Imminent integer overflow halted evaluation of the slope function at slope -4803063/178912 while computing a fixed point excluded interval about 55314/1484957. Imminent integer overflow halted evaluation of the slope function at slope -3492202/130083 while computing a fixed point excluded interval about 55314/1484957. Imminent integer overflow halted evaluation of the slope function at slope -5417540/201801 while computing a fixed point excluded interval about 55314/1484957. Imminent integer overflow halted evaluation of the slope function at slope -7598881/283055 while computing a fixed point excluded interval about 55314/1484957. Imminent integer overflow halted evaluation of the slope function at slope -7947073/296025 while computing a fixed point excluded interval about 55314/1484957. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 55314/1484957. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 55314/1484957. Imminent integer overflow halted evaluation of the slope function at slope -8735646/325399 while computing a fixed point excluded interval about 55314/1484957. Imminent integer overflow halted evaluation of the slope function at slope -8039262/299459 while computing a fixed point excluded interval about 55314/1484957. Imminent integer overflow halted evaluation of the slope function at slope -10046507/374228 while computing a fixed point excluded interval about 55314/1484957. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 55314/1484957. Imminent integer overflow halted evaluation of the slope function at slope -2703629/100709 while computing a fixed point excluded interval about 55314/1484957. Imminent integer overflow halted evaluation of the slope function at slope -4188586/156023 while computing a fixed point excluded interval about 55314/1484957. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 55314/1484957. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 55314/1484957. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 55314/1484957. Imminent integer overflow halted evaluation of the slope function at slope -11705560/436027 while computing a fixed point excluded interval about 55314/1484957. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 55314/1484957.