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 0/7, 0/35, 1/35, 1/7, 1/5, 2/7, 2/5, 3/5, 1/1, 2/1, 3/1, 4/1, 5/1, 6/1, 8/1 9/1, 10/1, 11/1, 12/1, 13/1, 15/1, 16/1, 22/1, 25/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,-1.000000) (-0.987465,-0.602083) (-0.598325,-0.585865) (-0.585749,-0.556729) (-0.556259,-0.520688) (-0.518346,-0.511697) (-0.510474,-0.509947) (-0.489979,-0.489606) (-0.489072,-0.488699) (-0.488372,-0.333333) (-0.329316,-0.017559) ( 0.017312,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 (-1.004829,-0.995171) -1/1 EXTENDED HST (-0.996400,-0.990630) -153/154 HST (-0.993491,-0.990891) -128/129 HST (-0.992188,-0.982856) -79/80 HST (-0.602843,-0.597045) -3/5 EXTENDED HST (-0.591564,-0.580567) -58/99 HST (-0.556785,-0.556424) -54/97 HST (-0.556773,-0.556014) -74/133 HST (-0.523434,-0.517133) -38/73 HST (-0.520073,-0.519928) -13/25 EXTENDED HST (-0.514965,-0.509003) -22/43 EXTENDED HST -> HST (-0.510969,-0.506905) -29/57 HST (-0.507747,-0.506031) -147/290 HST (-0.506851,-0.506847) -37/73 EXTENDED HST (-0.507226,-0.504659) -42/83 HST (-0.505489,-0.503630) -55/109 HST (-0.505300,-0.501805) -70/139 HST (-0.503546,-0.496503) -1/2 EXTENDED HST (-0.496974,-0.495785) -69/139 HST (-0.496403,-0.489362) -34/69 HST (-0.492540,-0.492535) -33/67 EXTENDED HST (-0.492396,-0.492217) -32/65 HST (-0.492220,-0.491901) -31/63 HST (-0.491859,-0.488155) -25/51 HST (-0.334759,-0.331920) -1/3 EXTENDED HST (-0.332482,-0.331115) -72/217 HST (-0.331780,-0.331140) -60/181 HST (-0.331466,-0.330694) -49/148 HST (-0.331037,-0.327571) -28/85 HST (-0.019923,-0.014496) -1/57 HST (-0.017107,-0.012281) -1/69 HST (-0.014256,-0.010134) -1/82 HST (-0.010634,-0.009500) -1/99 HST (-0.011029,-0.007839) -1/106 HST (-0.009085,-0.006526) -1/128 HST (-0.007675,-0.005366) -2/307 HST (-0.006512,-0.006475) -1/154 HST (-0.006196,-0.004436) -1/187 HST (-0.004783,0.004877 ) 0/1 EXTENDED HST ( 0.004328,0.006089 ) 1/191 HST ( 0.005253,0.005274 ) 1/190 HST ( 0.005273,0.005309 ) 1/189 HST ( 0.005315,0.007510 ) 1/156 HST ( 0.006429,0.006474 ) 1/155 HST ( 0.006492,0.009133 ) 1/129 HST ( 0.007790,0.011099 ) 1/107 HST ( 0.009371,0.013387 ) 1/88 HST ( 0.011487,0.016313 ) 1/71 HST ( 0.014085,0.014493 ) 1/70 HST ( 0.014327,0.020156 ) 1/58 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 2/1 1 35 No No Yes Yes NUMBER OF EQUATORS: 0 0 1 1 There are no more slope function fixed points. Number of excluded intervals computed by the fixed point finder: 18686 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,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,c,c,c,c,c,c,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,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,c,c,c,c,c,c,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");