These Thurston maps are NET maps for every choice of translation term. They are primitive and have degree 28. 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 0/2, 0/14, 1/14, 1/2, 2/2, 3/2, 4/2, 5/2, 6/2, 7/2 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,-0.126594) (-0.122202,-0.120238) (-0.119810,-0.107333) (-0.106759,-0.083454) (-0.082080,-0.081190) (-0.079507,-0.078396) (-0.072819,-0.070091) (-0.065599,-0.064841) (-0.060322,-0.059681) (-0.056393,-0.054743) (-0.046014,-0.044909) (-0.038861,-0.038070) (-0.033633,-0.033039) (-0.029645,-0.029182) (-0.026502,-0.026132) (-0.023962,-0.023659) (-0.021866,-0.021614) (-0.020107,-0.019894) ( 0.019894,0.020107 ) ( 0.021614,0.021866 ) ( 0.023659,0.023962 ) ( 0.026132,0.026502 ) ( 0.029182,0.029645 ) ( 0.033039,0.033633 ) ( 0.038070,0.038861 ) ( 0.044909,0.046014 ) ( 0.054743,0.056393 ) ( 0.059681,0.060322 ) ( 0.064841,0.065599 ) ( 0.070091,0.072819 ) ( 0.078396,0.079507 ) ( 0.081190,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 (-0.129847,-0.122831) -10/79 HST (-0.125149,-0.124851) -1/8 EXTENDED HST (-0.132988,-0.108978) -6/49 EXTENDED HST -> HST (-0.107371,-0.107240) -22/205 HST (-0.107315,-0.107162) -40/373 HST (-0.107216,-0.107070) -3/28 EXTENDED HST (-0.108756,-0.104375) -11/103 HST (-0.083468,-0.083436) -58/695 HST (-0.083463,-0.083408) -68/815 HST (-0.083590,-0.083227) -93/1115 HST (-0.083400,-0.083267) -1/12 EXTENDED HST (-0.089906,-0.072287) -6/73 HST (-0.074326,-0.065397) -2/29 EXTENDED HST -> HST (-0.066752,-0.066582) -1/15 EXTENDED HST (-0.069183,-0.061383) -2/31 EXTENDED HST -> HST (-0.062537,-0.062463) -1/16 EXTENDED HST (-0.066920,-0.054314) -3/49 EXTENDED HST -> HST (-0.060641,-0.060571) -2/33 EXTENDED HST (-0.058077,-0.051021) -2/37 EXTENDED HST -> HST (-0.052685,-0.052579) -1/19 EXTENDED HST (-0.055222,-0.044930) -3/59 HST (-0.050024,-0.049976) -1/20 EXTENDED HST (-0.047658,-0.041827) -2/45 EXTENDED HST -> HST (-0.043514,-0.043442) -1/23 EXTENDED HST (-0.041851,-0.041773) -11/263 HST (-0.041805,-0.041746) -17/407 HST (-0.043257,-0.040101) -22/527 HST (-0.041683,-0.041650) -1/24 EXTENDED HST (-0.040127,-0.040073) -16/399 HST (-0.040088,-0.040051) -22/549 HST (-0.040355,-0.039740) -32/799 HST (-0.040031,-0.039969) -1/25 EXTENDED HST (-0.039767,-0.039599) -6/151 HST (-0.039709,-0.039499) -4/101 HST (-0.039598,-0.039057) -2/51 HST (-0.040944,-0.036383) -3/77 HST (-0.038839,-0.034227) -2/55 HST (-0.035726,-0.035702) -1/28 EXTENDED HST (-0.034266,-0.034110) -4/117 HST (-0.034184,-0.033780) -2/59 HST (-0.035476,-0.031453) -3/89 HST (-0.031562,-0.031430) -5/159 HST (-0.034590,-0.027217) -6/191 HST (-0.031259,-0.031241) -1/32 EXTENDED HST (-0.028805,-0.025374) -4/147 HST (-0.027041,-0.027013) -1/37 EXTENDED HST (-0.026998,-0.023729) -2/79 HST (-0.025006,-0.024994) -1/40 EXTENDED HST (-0.025123,-0.022002) -2/85 HST (-0.023266,-0.023245) -1/43 EXTENDED HST (-0.022098,-0.021928) -2/91 HST (-0.023124,-0.020397) -3/137 HST (-0.022593,-0.017366) -7/344 HST (-0.020347,-0.020303) -6/295 HST (-0.020332,-0.020277) -4/197 HST (-0.020303,-0.020160) -2/99 HST (-0.033937,0.037786 ) 0/1 EXTENDED HST ( 0.035057,0.040798 ) 3/79 HST ( 0.040526,0.041065 ) 45/1103 HST ( 0.040800,0.040832 ) 2/49 EXTENDED HST ( 0.035508,0.045792 ) 3/73 HST ( 0.041650,0.041683 ) 1/24 EXTENDED HST ( 0.043442,0.043514 ) 1/23 EXTENDED HST ( 0.042580,0.048962 ) 3/65 HST ( 0.046289,0.047051 ) 2/43 EXTENDED HST -> HST ( 0.047576,0.047662 ) 1/21 EXTENDED HST ( 0.042411,0.054926 ) 3/61 HST ( 0.049976,0.050024 ) 1/20 EXTENDED HST ( 0.052579,0.052685 ) 1/19 EXTENDED HST ( 0.052147,0.060025 ) 3/53 HST ( 0.056807,0.057959 ) 2/35 EXTENDED HST -> HST ( 0.058758,0.058890 ) 1/17 EXTENDED HST ( 0.053357,0.066090 ) 7/116 HST ( 0.060571,0.060641 ) 2/33 EXTENDED HST ( 0.062463,0.062537 ) 1/16 EXTENDED HST ( 0.063522,0.068564 ) 8/121 HST ( 0.066582,0.066752 ) 1/15 EXTENDED HST ( 0.064104,0.075651 ) 2/29 EXTENDED HST -> HST ( 0.075452,0.076066 ) 5/66 HST ( 0.063499,0.085937 ) 7/92 HST ( 0.076810,0.077036 ) 1/13 EXTENDED HST The supplemental half-space computation shows that these NET maps are rational. SLOPE FUNCTION INFORMATION NUMBER OF FIXED POINTS: 1 EQUATOR? FIXED POINT c d 0 lambda1 lambda2 lambda1+lambda2 1/0 1 2 No No No No NUMBER OF EQUATORS: 0 0 0 0 There are no more slope function fixed points. Number of excluded intervals computed by the fixed point finder: 13645 There are no equators because both elementary divisors are greater than 1. 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=(2,26)(3,27)(4,24)(5,25)(6,22)(7,23)(8,20)(9,21)(10,18)(11,19)(12,16)(13,17)", "b=(1,27)(2,28)(3,25)(4,26)(5,23)(6,24)(7,21)(8,22)(9,19)(10,20)(11,17)(12,18)(13,15)(14,16)", "c=<1,1,1,1,c,c^-1,c,c^-1,c,c^-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)", "d=(1,28)(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=(1,3)(2,28)(4,26)(5,27)(6,24)(7,25)(8,22)(9,23)(10,20)(11,21)(12,18)(13,19)(14,16)(15,17)", "b=(3,27)(4,28)(5,25)(6,26)(7,23)(8,24)(9,21)(10,22)(11,19)(12,20)(13,17)(14,18)", "c=<1,d,1,1,1,1,c,c^-1,c,c^-1,c,c^-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)", "d=(1,28)(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=<1,c^-1,1,1,c,1,c,c^-1,1,c^-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,c*d,1>(1,4)(2,27)(3,6)(5,8)(7,10)(9,12)(11,14)(13,16)(15,18)(17,20)(19,22)(21,24)(23,26)(25,28)", "b=<1,1,1,1,c,c^-1,c,c^-1,c,c^-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)", "c=(1,27)(2,28)(3,25)(4,26)(5,23)(6,24)(7,21)(8,22)(9,19)(10,20)(11,17)(12,18)(13,15)(14,16)", "d=(1,25)(3,23)(4,28)(5,21)(6,26)(7,19)(8,24)(9,17)(10,22)(11,15)(12,20)(14,18)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=<1,c^-1,1,1,1,1,c,1,c,c^-1,1,c^-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,c,1>(1,4)(2,27)(3,6)(5,8)(7,10)(9,12)(11,14)(13,16)(15,18)(17,20)(19,22)(21,24)(23,26)(25,28)", "b=<1,d,1,1,1,1,c,c^-1,c,c^-1,c,c^-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)", "c=(3,27)(4,28)(5,25)(6,26)(7,23)(8,24)(9,21)(10,22)(11,19)(12,20)(13,17)(14,18)", "d=(1,27)(2,4)(3,25)(5,23)(6,28)(7,21)(8,26)(9,19)(10,24)(11,17)(12,22)(13,15)(14,20)(16,18)", "a*b*c*d");