These Thurston maps are NET maps for every choice of translation term. They are primitive and have degree 39. 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 7. ALL THURSTON MULTIPLIERS c/d IN UNREDUCED FORM 1/39, 1/13, 1/3, 1/1, 3/3, 5/3, 7/3, 9/3, 11/3, 5/1, 7/1, 11/1, 17/1, 19/1 23/1, 25/1, 29/1, 31/1, 35/1, 37/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,0.027682) ( 0.027967,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 (0.027639,0.027722) 9/325 HST (0.027665,0.027749) 14/505 HST (0.027741,0.027756) 27/973 HST (0.027751,0.027761) 36/1297 HST (0.027744,0.027772) 46/1657 HST (0.027772,0.027783) 1/36 EXTENDED HST (0.027778,0.027820) 19/683 HST (0.027819,0.027820) 37/1330 HST (0.027820,0.027820) 257/9238 HST (0.027820,0.027820) 55/1977 HST (0.027820,0.027820) 73/2624 HST (0.027820,0.027821) 18/647 HST (0.027821,0.027821) 125/4493 HST (0.027821,0.027821) 71/2552 HST (0.027821,0.027822) 53/1905 HST (0.027822,0.027822) 35/1258 HST (0.027822,0.027823) 52/1869 HST (0.027822,0.027823) 69/2480 HST (0.027822,0.027823) 86/3091 HST (0.027823,0.027824) 17/611 HST (0.027824,0.027824) 84/3019 HST (0.027824,0.027826) 33/1186 HST (0.027825,0.027827) 16/575 HST (0.027827,0.027833) 15/539 HST (0.027829,0.027839) 14/503 HST (0.027834,0.027843) 13/467 HST (0.027837,0.027850) 12/431 HST (0.027848,0.027874) 10/359 HST (0.027860,0.028058) 7/251 HST (0.027892,0.028087) 5/179 HST The supplemental half-space computation shows that these NET maps are rational. SLOPE FUNCTION INFORMATION NUMBER OF FIXED POINTS FOUND: 4 EQUATOR? FIXED POINT c d 0 lambda1 lambda2 lambda1+lambda2 -36/1 1 39 Yes Yes No No 0/1 1 39 Yes Yes No No -10962/305 1 39 Yes Yes No No -4564/127 1 39 Yes Yes No No NUMBER OF EQUATORS FOUND: 4 4 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: 1753 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,0.027822) ( 0.027822,0.027822) ( 0.027822,0.027822) ( 0.027822,0.027823) ( 0.027823,0.027823) ( 0.027823,0.027823) ( 0.027823,0.027824) ( 0.027824,0.027824) ( 0.027824,0.027832) ( 0.027832,0.027832) ( 0.027832,0.027832) ( 0.027832,0.027832) ( 0.027832,0.027832) ( 0.027832,0.027859) ( 0.027859,0.027859) ( 0.027859,infinity) NONTRIVIAL CYCLES -611/17 -> -647/18 -> -611/17 -1869/52 -> -1905/53 -> -1869/52 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 1 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. 40/39 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,a*b,b,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,b^-1,b^-1>(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)", "b=(1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)", "c=(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)", "d=(1,2)(3,39)(4,38)(5,37)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=(1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)", "b=(1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)", "c=(1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)", "d=(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)", "b=(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)", "c=(1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)", "d=(1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)", "b=(1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)", "c=(1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)", "d=(1,37)(2,36)(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)(38,39)", "a*b*c*d"); ****************************INTEGER OVERFLOW REPORT***************************** Imminent integer overflow halted evaluation of the slope function at slope -897382/24967 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -1463749/40725 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -874008/24317 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -3074057/85529 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -861127/23959 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -2526665/70299 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -14546430/404731 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -1858784/51719 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -1835998/51085 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -4707656/131023 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -4700470/130823 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -1066043/29670 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -4139962/115223 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -1531300/42619 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -7969888/222033 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -7958258/221709 during the search for all slope function fixed points.