These Thurston maps are NET maps for every choice of translation term. They are primitive and have degree 35. 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 0/35, 1/35, 1/7, 1/5, 2/7, 2/5, 3/7, 3/5, 4/5, 1/1, 5/5, 2/1, 3/1, 4/1, 6/1 8/1, 9/1, 11/1, 12/1, 13/1, 16/1, 17/1, 23/1, 26/1, 29/1, 32/1, 33/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,-1.000000) (-0.987653,-0.500000) (-0.494410,-0.337272) (-0.328384,-0.017312) ( 0.016087,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 (-1.007754,-0.992479) -1/1 EXTENDED HST (-0.995597,-0.988555) -126/127 HST (-0.992113,-0.983016) -80/81 HST (-0.503546,-0.496503) -1/2 EXTENDED HST (-0.497489,-0.494538) -63/127 HST (-0.496030,-0.493771) -48/97 HST (-0.494795,-0.494677) -47/95 HST (-0.494626,-0.494622) -46/93 EXTENDED HST (-0.494581,-0.494428) -45/91 HST (-0.337900,-0.336529) -29/86 HST (-0.337612,-0.335174) -36/107 HST (-0.336010,-0.334336) -61/182 HST (-0.334538,-0.332138) -1/3 EXTENDED HST (-0.333333,-0.323944) -23/70 HST (-0.019585,-0.014757) -1/58 HST (-0.017107,-0.012281) -1/68 HST (-0.014435,-0.010010) -1/82 HST (-0.010038,-0.009962) -1/100 HST (-0.011238,-0.008312) -1/101 HST (-0.009582,-0.006811) -1/121 HST (-0.006844,-0.006762) -1/147 HST (-0.007881,-0.005628) -1/148 HST (-0.006618,-0.004633) -1/178 HST (-0.004783,0.004877 ) 0/1 EXTENDED HST ( 0.004869,0.006744 ) 1/172 HST ( 0.005838,0.008252 ) 1/141 HST ( 0.007122,0.010137 ) 1/116 HST ( 0.008656,0.012177 ) 1/96 HST ( 0.010467,0.010586 ) 1/95 HST ( 0.010607,0.015053 ) 1/78 HST ( 0.012848,0.018461 ) 1/64 HST ( 0.015651,0.016102 ) 1/63 HST The supplemental half-space computation shows that these NET maps are rational. SLOPE FUNCTION INFORMATION NUMBER OF FIXED POINTS FOUND: 18 EQUATOR? FIXED POINT c d 0 lambda1 lambda2 lambda1+lambda2 2/1 1 35 No No Yes Yes 34/1 1 35 No No Yes Yes 34/3 1 35 No No Yes Yes 34/5 1 35 No No Yes Yes 34/7 1 35 No No Yes Yes 34/9 1 35 No No Yes Yes 32/11 1 35 No No Yes Yes 34/11 1 35 No No Yes Yes 34/13 1 35 No No Yes Yes 34/15 1 35 No No Yes Yes 34/19 1 35 No No Yes Yes 34/21 1 35 No No Yes Yes 34/23 1 35 No No Yes Yes 34/25 1 35 No No Yes Yes 34/27 1 35 No No Yes Yes 34/29 1 35 No No Yes Yes 34/31 1 35 No No Yes Yes 34/33 1 35 No No Yes Yes NUMBER OF EQUATORS FOUND: 0 0 18 18 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: 10446 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,-0.484821) (-0.484821,-0.484821) (-0.484821,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=(1,34)(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=(1,34)(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=(1,35)(2,34)(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)", "d=(1,35)(2,34)(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=(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)", "b=<1,b,c^-1*b,c^-1*b,c^-1*b,c^-1*b,c^-1*b,c^-1*b,c^-1*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*c,b^-1*c,b^-1*c,b^-1*c,b^-1*c,b^-1*c,b^-1*c,b^-1*c>(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)", "c=(1,35)(2,34)(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)", "d=(1,35)(2,34)(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 lambda2: NewSphereMachine( "a=(1,35)(2,34)(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)", "b=(1,35)(2,34)(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)", "c=(1,34)(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=(1,34)(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 lambda1+lambda2: NewSphereMachine( "a=(1,35)(2,34)(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)", "b=(1,35)(2,34)(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)", "c=<1,b,c^-1*b,c^-1*b,c^-1*b,c^-1*b,c^-1*b,c^-1*b,c^-1*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*c,b^-1*c,b^-1*c,b^-1*c,b^-1*c,b^-1*c,b^-1*c,b^-1*c>(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)", "d=(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)", "a*b*c*d"); ****************************INTEGER OVERFLOW REPORT***************************** Imminent integer overflow occurred during the computation of a fixed point excluded interval about -28347/58469. Imminent integer overflow halted evaluation of the slope function at slope 4312410/2090747 while computing a fixed point excluded interval about -1030665/2125867. Imminent integer overflow occurred during the computation of a fixed point excluded interval about -118738/244911. Imminent integer overflow occurred during the computation of a fixed point excluded interval about -118738/244911. Imminent integer overflow occurred during the computation of a fixed point excluded interval about -118738/244911. Imminent integer overflow occurred during the computation of a fixed point excluded interval about -265823/548291. Imminent integer overflow occurred during the computation of a fixed point excluded interval about -265823/548291. Imminent integer overflow occurred during the computation of a fixed point excluded interval about -263683/543877. Imminent integer overflow occurred during the computation of a fixed point excluded interval about -263683/543877. Imminent integer overflow occurred during the computation of a fixed point excluded interval about -881440/1818073. Imminent integer overflow occurred during the computation of a fixed point excluded interval about -881440/1818073. Imminent integer overflow occurred during the computation of a fixed point excluded interval about -365285/753443. Imminent integer overflow occurred during the computation of a fixed point excluded interval about -365285/753443. Imminent integer overflow occurred during the computation of a fixed point excluded interval about -365285/753443. Imminent integer overflow halted evaluation of the slope function at slope 5474872/2654333 while computing a fixed point excluded interval about -2289048/4721429. Imminent integer overflow halted evaluation of the slope function at slope 5073330/2459657 during the search for all slope function fixed points. Imminent integer overflow halted computation of the slope function at slope -1379830/991241 during the modular group computation. This caused the modular group computation to abort.