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 10. 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 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,0.031117) ( 0.031533,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 (0.031034,0.031164) 22/707 HST (0.031118,0.031138) 8/257 HST (0.031145,0.031189) 12/385 HST (0.031180,0.031199) 16/513 HST (0.031173,0.031218) 20/641 HST (0.031211,0.031225) 31/993 HST (0.031219,0.031231) 40/1281 HST (0.031227,0.031235) 52/1665 HST (0.031232,0.031238) 66/2113 HST (0.031236,0.031240) 82/2625 HST (0.031238,0.031242) 100/3201 HST (0.031232,0.031250) 123/3937 HST (0.031245,0.031255) 1/32 EXTENDED HST (0.031250,0.031309) 17/543 HST (0.031309,0.031310) 33/1054 HST (0.031309,0.031310) 49/1565 HST (0.031310,0.031312) 16/511 HST (0.031312,0.031313) 63/2012 HST (0.031312,0.031313) 47/1501 HST (0.031312,0.031313) 78/2491 HST (0.031313,0.031313) 31/990 HST (0.031313,0.031314) 77/2459 HST (0.031314,0.031314) 46/1469 HST (0.031313,0.031315) 107/3417 HST (0.031314,0.031314) 61/1948 HST (0.031314,0.031316) 15/479 HST (0.031316,0.031316) 401/12805 HST (0.031316,0.031317) 59/1884 HST (0.031317,0.031318) 29/926 HST (0.031318,0.031319) 57/1820 HST (0.031319,0.031321) 14/447 HST (0.031320,0.031325) 27/862 HST (0.031323,0.031329) 13/415 HST (0.031327,0.031333) 37/1181 HST (0.031332,0.031352) 11/351 HST (0.031339,0.031359) 10/319 HST (0.031354,0.031367) 9/287 HST (0.031360,0.031610) 7/223 HST (0.031396,0.031447) 6/191 HST (0.031435,0.031707) 4/127 HST The supplemental half-space computation shows that these NET maps are rational. SLOPE FUNCTION INFORMATION NUMBER OF FIXED POINTS FOUND: 3 EQUATOR? FIXED POINT c d 0 lambda1 lambda2 lambda1+lambda2 -32/1 1 35 Yes Yes No No 0/1 1 35 Yes Yes No No -9802/307 1 35 Yes Yes No No NUMBER OF EQUATORS FOUND: 3 3 0 0 The fixed point finder is unable to determine whether there are any more slope function fixed points. Number of excluded intervals computed by the fixed point finder: 1931 Here is their union. There are no more slope function fixed points whose negative reciprocals lie in any of the following intervals. EXCLUDED INTERVALS FOR THE FIXED POINT COMPUTATION (-infinity,0.031193) ( 0.031193,0.031279) ( 0.031279,0.031279) ( 0.031279,0.031336) ( 0.031336,0.031336) ( 0.031336,infinity) NONTRIVIAL CYCLES -447/14 -> -479/15 -> -447/14 The slope function maps every slope to a slope: no slope maps to the nonslope. The slope function orbit of every slope p/q with |p| <= 50 and |q| <= 50 ends in one of the above cycles. FUNDAMENTAL GROUP WREATH RECURSIONS When the translation term of the affine map is 0: NewSphereMachine( "a=<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)", "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,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c*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)", "d=<1,1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c>(1,2)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)", "a*b*c*d"); When the translation term of the affine map is lambda1: 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=(1,34)(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,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c*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"); 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,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c^-1,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c*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)", "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=(1,34)(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=(1,34)(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=(1,35)(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=(1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,35)", "a*b*c*d"); ****************************INTEGER OVERFLOW REPORT***************************** Imminent integer overflow halted evaluation of the slope function at slope -351582288/10966751 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -1255518/39271 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -1257692/39339 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -5813062/182075 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -1189899/37280 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -6126122/191969 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -4657530/145949 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -7104928/222641 during the search for all slope function fixed points.