These Thurston maps are NET maps for every choice of translation term. They are primitive and have degree 31. 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 1/31, 1/1, 3/1, 5/1, 7/1, 9/1, 11/1, 13/1, 15/1, 17/1, 19/1, 21/1, 23/1 25/1, 27/1, 29/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,0.035542) ( 0.036092,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 (0.035493,0.035602) 8/225 HST (0.035565,0.035640) 12/337 HST (0.035618,0.035663) 18/505 HST (0.035649,0.035676) 25/701 HST (0.035667,0.035686) 34/953 HST (0.035679,0.035692) 45/1261 HST (0.035688,0.035697) 58/1625 HST (0.035693,0.035700) 73/2045 HST (0.035698,0.035703) 93/2605 HST (0.035701,0.035705) 113/3165 HST (0.035704,0.035707) 141/3949 HST (0.035707,0.035722) 1/28 EXTENDED HST (0.035714,0.035802) 15/419 HST (0.035801,0.035803) 29/810 HST (0.035803,0.035804) 43/1201 HST (0.035803,0.035805) 57/1592 HST (0.035804,0.035807) 14/391 HST (0.035806,0.035808) 55/1536 HST (0.035808,0.035810) 27/754 HST (0.035809,0.035811) 40/1117 HST (0.035811,0.035812) 79/2206 HST (0.035812,0.035814) 13/363 HST (0.035814,0.035815) 51/1424 HST (0.035815,0.035818) 25/698 HST (0.035815,0.035823) 37/1033 HST (0.035819,0.035822) 12/335 HST (0.035818,0.035830) 59/1647 HST (0.035823,0.035844) 11/307 HST (0.035840,0.035844) 10/279 HST (0.035835,0.035858) 49/1367 HST (0.035845,0.035876) 9/251 HST (0.035866,0.035889) 8/223 HST (0.035870,0.035911) 22/613 HST (0.035896,0.036002) 6/167 HST (0.035968,0.036239) 4/111 HST The supplemental half-space computation shows that these NET maps are rational. SLOPE FUNCTION INFORMATION NUMBER OF FIXED POINTS FOUND: 5 EQUATOR? FIXED POINT c d 0 lambda1 lambda2 lambda1+lambda2 -28/1 1 31 Yes Yes No No 0/1 1 31 Yes Yes No No -5054/181 1 31 Yes Yes No No -6728/241 1 31 Yes Yes No No -6724/241 1 31 Yes Yes No No NUMBER OF EQUATORS FOUND: 5 5 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: 1060 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,0.035813) ( 0.035813,0.035884) ( 0.035884,0.035884) ( 0.035884,infinity) NONTRIVIAL CYCLES -363/13 -> -391/14 -> -363/13 -642/23 -> -698/25 -> -642/23 -1033/37 -> -1229/44 -> -1145/41 -> -1117/40 -> -1033/37 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^-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>(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)", "b=(1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)", "c=(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)", "d=(1,2)(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)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=(1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)", "b=(1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)", "c=(1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)", "d=(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)", "b=(2,31)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)", "c=(1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)", "d=(1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)", "b=(1,31)(2,30)(3,29)(4,28)(5,27)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)", "c=(1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)", "d=(1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(30,31)", "a*b*c*d"); ****************************INTEGER OVERFLOW REPORT***************************** Imminent integer overflow halted evaluation of the slope function at slope -4078466/146063 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -2069904/74189 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -2301536/82589 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -2296074/82393 during the search for all slope function fixed points.