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 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 0/4, 0/14, 0/28, 1/7, 1/4, 1/2, 2/4, 1/1, 2/2, 3/2, 2/1, 5/2, 3/1, 4/1, 5/1 6/1, 7/1, 11/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,-0.046512) (-0.042333,-0.041021) (-0.037037,-0.034483) (-0.031624,-0.030885) (-0.028986,-0.028169) (-0.028073,-0.027489) (-0.025238,-0.024766) (-0.024217,-0.023415) (-0.022924,-0.022534) (-0.020999,-0.020671) (-0.020619,-0.020202) ( 0.020202,0.020619 ) ( 0.020671,0.020999 ) ( 0.022534,0.022924 ) ( 0.023415,0.024217 ) ( 0.024766,0.025238 ) ( 0.027489,0.028073 ) ( 0.028169,0.028986 ) ( 0.030885,0.031624 ) ( 0.034483,0.037037 ) ( 0.041021,0.042333 ) ( 0.046512,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 (-0.046526,-0.046497) -2/43 EXTENDED HST (-0.051153,-0.040978) -3/65 HST (-0.045528,-0.045381) -1/22 EXTENDED HST (-0.044530,-0.031574) -1/27 EXTENDED HST -> HST (-0.031000,-0.030234) -2/65 HST (-0.030315,-0.030291) -1/33 EXTENDED HST (-0.035205,-0.024891) -2/67 HST (-0.029443,-0.029381) -1/34 EXTENDED HST (-0.029366,-0.020363) -2/81 HST (-0.024405,-0.024375) -1/41 EXTENDED HST (-0.020205,-0.020199) -2/99 EXTENDED HST (-0.022311,-0.017923) -3/149 HST (-0.020014,-0.019986) -1/50 EXTENDED HST (-0.018182,-0.017544) -1/56 HST (-0.020440,-0.014211) -1/57 HST (-0.014431,-0.014143) -2/141 HST (-0.014176,-0.014126) -4/283 HST (-0.014140,-0.014118) -5/354 HST (-0.014124,-0.014102) -6/425 HST (-0.014109,-0.014096) -12/851 HST (-0.014660,-0.013530) -18/1277 HST (-0.014090,-0.014079) -1/71 EXTENDED HST (-0.013971,-0.013088) -11/813 HST (-0.013525,-0.013502) -1/74 EXTENDED HST (-0.014172,-0.011989) -7/535 HST (-0.013073,-0.013071) -2/153 EXTENDED HST (-0.013072,-0.012903) -1/77 HST (-0.012048,-0.011765) -1/84 HST (-0.013752,-0.009581) -1/85 HST (-0.010374,-0.008774) -5/522 HST (-0.009570,-0.009569) -2/209 EXTENDED HST (-0.009569,-0.009479) -1/105 HST (-0.010808,-0.006685) -1/114 HST (-0.008418,0.008418 ) 0/1 EXTENDED HST ( 0.007989,0.012917 ) 1/95 HST ( 0.010566,0.017083 ) 1/73 HST ( 0.013816,0.013962 ) 1/72 HST ( 0.014079,0.014090 ) 1/71 EXTENDED HST ( 0.014242,0.023026 ) 1/53 HST ( 0.019092,0.019372 ) 1/52 HST ( 0.019603,0.019613 ) 1/51 EXTENDED HST ( 0.019986,0.020014 ) 1/50 EXTENDED HST ( 0.022989,0.023095 ) 3/130 HST ( 0.019614,0.026695 ) 4/173 HST ( 0.023242,0.023269 ) 1/43 EXTENDED HST ( 0.026378,0.027206 ) 1/37 EXTENDED HST -> HST ( 0.018785,0.036160 ) 2/73 HST ( 0.031574,0.044530 ) 1/27 EXTENDED HST -> HST ( 0.043769,0.045552 ) 5/112 HST ( 0.044763,0.044981 ) 3/67 HST ( 0.045381,0.045528 ) 1/22 EXTENDED HST ( 0.040978,0.051153 ) 4/87 HST ( 0.046135,0.046459 ) 3/65 HST ( 0.046497,0.046526 ) 2/43 EXTENDED HST The supplemental half-space computation shows that these NET maps are rational. SLOPE FUNCTION INFORMATION There are no slope function fixed points because every loop multiplier of the mod 2 slope correspondence graph is at least 1 and the map is rational. 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 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,d*b*c,d*a,c^-1*b*c,d*a*c,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(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)", "b=<1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(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)", "c=<1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(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=(1,2)(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)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=<1,b*c,d*a,b,b^-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(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)", "b=<1,1,d,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(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)", "c=(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=(1,2)(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)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=<1,d*a*d^-1,b*c,d*a*c,c^-1*b*c,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(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)", "b=<1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(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=<1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(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)", "d=<1,d*a,d*b*c,d*a*c,c^-1*b*c,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(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)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=<1,1,b,b^-1,b,b^-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(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)", "b=(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=<1,1,d,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(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)", "d=<1,a,a^-1,b^-1,b,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(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)", "a*b*c*d");