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, 8/2 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,-0.107333) (-0.106765,-0.103803) (-0.103272,-0.075374) (-0.074815,-0.073348) (-0.072819,-0.057913) (-0.057583,-0.056710) (-0.056393,-0.054743) (-0.054447,-0.053666) (-0.051636,-0.050933) (-0.049101,-0.048465) (-0.048233,-0.047021) (-0.046803,-0.046224) (-0.046014,-0.044909) (-0.044710,-0.044182) (-0.042797,-0.042313) (-0.041040,-0.040595) (-0.040432,-0.039577) (-0.038861,-0.038070) (-0.034804,-0.034168) (-0.033633,-0.033039) (-0.030550,-0.030060) (-0.029645,-0.029182) (-0.027224,-0.026833) (-0.026502,-0.026132) (-0.024550,-0.024232) (-0.023962,-0.023659) (-0.022355,-0.022091) (-0.021866,-0.021614) (-0.020520,-0.020297) (-0.020107,-0.019894) ( 0.019894,0.020107 ) ( 0.021156,0.021398 ) ( 0.021614,0.021866 ) ( 0.023112,0.023401 ) ( 0.023659,0.023962 ) ( 0.025467,0.025818 ) ( 0.026132,0.026502 ) ( 0.028355,0.028791 ) ( 0.029182,0.029645 ) ( 0.031982,0.032539 ) ( 0.033039,0.033633 ) ( 0.036674,0.037407 ) ( 0.038070,0.038861 ) ( 0.040595,0.041040 ) ( 0.042313,0.042797 ) ( 0.042979,0.043989 ) ( 0.044182,0.044710 ) ( 0.044909,0.046014 ) ( 0.046224,0.046803 ) ( 0.048465,0.049101 ) ( 0.050933,0.051636 ) ( 0.051902,0.053382 ) ( 0.053608,0.067876 ) ( 0.068336,0.069607 ) ( 0.070091,0.093605 ) ( 0.094041,0.096466 ) ( 0.096929,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 (-0.108370,-0.106233) -22/205 HST (-0.107216,-0.107070) -3/28 EXTENDED HST (-0.105880,-0.101547) -11/106 HST (-0.103516,-0.103380) -3/29 EXTENDED HST (-0.080974,-0.068405) -7/93 HST (-0.075036,-0.074964) -3/40 EXTENDED HST (-0.062298,-0.052513) -7/121 HST (-0.057713,-0.057671) -3/52 EXTENDED HST (-0.055563,-0.047062) -3/58 HST (-0.050622,-0.042613) -7/149 HST (-0.046889,-0.046861) -3/64 EXTENDED HST (-0.045500,-0.038332) -3/71 HST (-0.041700,-0.041634) -1/24 EXTENDED HST (-0.038104,-0.037847) -3/79 HST (-0.037927,-0.037547) -2/53 HST (-0.039142,-0.035533) -3/80 HST (-0.037063,-0.037011) -1/27 EXTENDED HST (-0.036859,-0.034007) -6/169 HST (-0.035482,-0.035315) -4/113 HST (-0.035302,-0.035286) -3/85 EXTENDED HST (-0.035253,-0.034924) -2/57 HST (-0.034053,-0.033745) -2/59 HST (-0.034505,-0.032940) -11/326 HST (-0.033715,-0.033701) -3/89 EXTENDED HST (-0.033646,-0.032184) -11/334 HST (-0.032931,-0.032644) -2/61 HST (-0.032278,-0.032238) -1/31 EXTENDED HST (-0.034376,-0.028944) -3/94 HST (-0.031881,-0.031612) -2/63 HST (-0.031269,-0.031231) -1/32 EXTENDED HST (-0.030138,-0.027328) -3/104 HST (-0.028587,-0.028556) -1/35 EXTENDED HST (-0.027339,-0.027207) -3/110 HST (-0.028933,-0.024293) -7/261 HST (-0.026790,-0.026781) -3/112 EXTENDED HST (-0.026762,-0.026572) -2/75 HST (-0.026134,-0.021936) -7/289 HST (-0.024197,-0.024190) -3/124 EXTENDED HST (-0.024174,-0.024019) -2/83 HST (-0.022043,-0.021914) -5/228 HST (-0.022416,-0.021383) -11/502 HST (-0.021901,-0.021895) -3/137 EXTENDED HST (-0.022447,-0.020166) -5/234 HST (-0.021285,-0.021268) -1/47 EXTENDED HST (-0.020257,-0.020148) -5/248 HST (-0.020611,-0.019658) -11/546 HST (-0.020137,-0.020132) -3/149 EXTENDED HST (-0.044001,0.050701 ) 0/1 EXTENDED HST ( 0.047805,0.053387 ) 7/138 HST ( 0.050831,0.050864 ) 3/59 EXTENDED HST ( 0.048136,0.057911 ) 7/131 HST ( 0.053553,0.053590 ) 3/56 EXTENDED HST ( 0.061161,0.073716 ) 7/103 HST ( 0.068152,0.068211 ) 3/44 EXTENDED HST ( 0.092287,0.094920 ) 22/235 HST ( 0.093694,0.093806 ) 3/32 EXTENDED HST ( 0.093781,0.099089 ) 11/114 HST ( 0.096715,0.096834 ) 3/31 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: 10924 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=<1,c^-1,b,c^-1,b,c^-1,b,c^-1,b,c^-1,b,1,1,1,1,1,1,c,1,c,b^-1,c,b^-1,c,b^-1,c,b^-1*c,1>(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,d,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=<1,1,b,c^-1,c^-1*b,c^-1,b,c^-1,b,c^-1,b,c^-1,1,1,1,1,1,1,1,c,b^-1,c,b^-1,c,b^-1*c,c,b^-1*c,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)", "c=<1,1,1,1,1,1,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,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,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,a*b,1,d,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*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=<1,1,1,1,1,1,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=<1,1,b,c^-1,c^-1*b,c^-1,b,c^-1,b,c^-1,b,c^-1,1,1,1,1,1,1,1,c,b^-1,c,b^-1,c,b^-1*c,c,b^-1*c,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");