These Thurston maps are NET maps for every choice of translation term. They are primitive and have degree 15. 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 0/5, 0/15, 1/15, 1/5, 1/3, 2/3, 1/1, 3/3, 2/1, 3/1, 4/1, 6/1, 7/1, 9/1, 12/1 13/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,0.000000) ( 0.000000,0.074074) ( 0.074300,0.075599) ( 0.080778,0.081773) ( 0.085874,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 (-0.040746,0.023507) 0/1 EXTENDED HST ( 0.073551,0.075395) 2/27 EXTENDED HST -> HST ( 0.075533,0.076008) 5/66 HST ( 0.075798,0.076065) 13/171 HST ( 0.076031,0.076137) 7/92 HST ( 0.076127,0.076424) 8/105 HST ( 0.076299,0.076526) 12/157 HST ( 0.076508,0.076564) 15/196 HST ( 0.076381,0.076708) 17/222 HST ( 0.076665,0.076773) 28/365 HST ( 0.076734,0.076810) 40/521 HST ( 0.076805,0.077033) 1/13 EXTENDED HST ( 0.077009,0.077070) 50/649 HST ( 0.077043,0.077045) 49/636 HST ( 0.077044,0.077121) 36/467 HST ( 0.077090,0.077095) 35/454 HST ( 0.077097,0.077225) 25/324 HST ( 0.077162,0.077179) 24/311 HST ( 0.077172,0.077316) 18/233 HST ( 0.077265,0.078221) 6/77 HST ( 0.078025,0.078245) 5/64 HST ( 0.078172,0.078828) 4/51 HST ( 0.078767,0.078947) 19/241 HST ( 0.078912,0.078984) 3/38 EXTENDED HST ( 0.078652,0.079365) 8/101 HST ( 0.079339,0.079790) 5/63 HST ( 0.079736,0.079857) 47/589 HST ( 0.079799,0.079824) 17/213 HST ( 0.079831,0.079893) 23/288 HST ( 0.079875,0.080136) 2/25 EXTENDED HST ( 0.080100,0.080408) 13/162 HST ( 0.080280,0.080492) 7/87 HST ( 0.080236,0.080737) 26/323 HST ( 0.080504,0.080773) 5/62 HST ( 0.080641,0.080901) 50/619 HST ( 0.081663,0.082409) 5/61 HST ( 0.082101,0.082674) 8/97 HST ( 0.082430,0.082986) 11/133 HST ( 0.082905,0.083090) 20/241 HST ( 0.083089,0.083091) 57/686 HST ( 0.082998,0.083197) 86/1035 HST ( 0.083094,0.083612) 1/12 EXTENDED HST ( 0.083333,0.085366) 6/71 HST ( 0.084520,0.086840) 4/47 EXTENDED HST -> HST ( 0.085361,0.087241) 3/35 EXTENDED HST -> HST The supplemental half-space computation shows that these NET maps are rational. SLOPE FUNCTION INFORMATION NUMBER OF FIXED POINTS: 8 EQUATOR? FIXED POINT c d 0 lambda1 lambda2 lambda1+lambda2 -12/1 1 15 Yes Yes No No 0/1 1 15 Yes Yes No No -38/3 1 15 Yes Yes No No -28/3 1 15 Yes Yes No No -112/9 1 15 Yes Yes No No -168/13 1 15 Yes Yes No No -140/11 1 15 Yes Yes No No -56/5 1 15 Yes Yes No No NUMBER OF EQUATORS: 8 8 0 0 There are no more slope function fixed points. Number of excluded intervals computed by the fixed point finder: 2159 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,b,b,b,b,b,b,b,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1>(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)", "b=(1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)", "c=<1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c,c,c,c,c,c,c*d>(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)", "d=<1,1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c,c,c,c,c,c,c>(1,2)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=(1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)", "b=(1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)", "c=(1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)", "d=<1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c,c,c,c,c,c,c*d>(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)", "b=<1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c,c,c,c,c,c,c*d>(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)", "c=(1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)", "d=(1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)", "b=(1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)", "c=(1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)", "d=(1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,15)", "a*b*c*d");