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 7. ALL THURSTON MULTIPLIERS c/d IN UNREDUCED FORM 1/35, 1/7, 1/5, 3/7, 3/5, 1/1, 5/5, 3/1, 9/1, 11/1, 13/1, 17/1, 19/1, 23/1 27/1, 29/1, 31/1, 33/1 Every NET map in these pure modular group Hurwitz classes is rational because the mod 2 slope correspondence graph has no loops. EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,-0.031491) (-0.031311,infinity ) SLOPE FUNCTION INFORMATION There are no slope function fixed points because the mod 2 slope correspondence graph has no loops. NONTRIVIAL CYCLES 3127/98 -> 31621/991 -> 72974/2287 -> 283217/8876 -> 63593/1993 -> 22240/697 -> 3127/98 19113/599 -> 60466/1895 -> 41353/1296 -> 152553/4781 -> 34748/1089 -> 47607/1492 -> 19113/599 The slope function maps every slope to a slope: no slope maps to the nonslope. There are 3096 slopes s = p/q with |p| <= 50 and |q| <= 50. The 100 slopes s in the following list have the property that the slope function orbit of s contains a slope t whose numerator or denominator exceeds 1,000,000 in absolute value, and the slopes between s and t are not among the slopes p/q with |p| <= 50 and |q| <= 50. -48/1, -46/1, -45/1, -43/1, -32/1, -30/1, -20/1, -16/1, -13/1, 6/1, 16/1, 25/1, 44/1, -41/2, 9/2, 15/2, 17/2, -44/3, -34/3, -31/3, -5/3, 20/3, 38/3, 50/3, 31/4, -17/5, 32/5, 34/5, 44/5, -19/6, 47/6, -31/7, -23/7, 18/7, 26/7, 44/7, 46/7, -29/9, -26/9, 26/9, 38/9, 41/10, 43/10, -34/11, 17/11, 41/12, -38/13, -21/13, 29/13, 46/13, 50/13, -47/14, 1/14, 17/15, 23/15, 38/15, 41/15, -43/16, 27/17, 46/17, -33/19, 29/19, -43/21, -49/23, -37/23, 29/23, -43/25, -23/25, 3/25, 13/25, 34/27, 41/27, 43/27, -49/29, -45/29, -34/29, -26/29, 7/29, 13/30, -15/31, 19/33, 41/33, -41/34, -17/35, 43/35, 19/36, -12/37, 23/37, -5/38, 23/38, 17/39, 13/40, 31/40, -9/43, 15/43, 23/43, 23/45, -32/47, -3/47, 23/48 There are 102 of these slopes. These are the first 100 of them. The slope function orbit of every slope p/q with |p| <= 50 and |q| <= 50 either contains an extended rational number whose numerator or denominator exceeds 1,000,000 in absolute value or ends in one of the above cycles. 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,a*b,b,b,b,b,b,b,b,b,b,b,b,b,b,b,b,b,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1>(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,a*b,b,b,b,b,b,b,b,b,b,b,b,b,b,b,b,b,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1>(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"); ****************************INTEGER OVERFLOW REPORT***************************** Imminent integer overflow caused the modular group computation to abort.