These Thurston maps are NET maps for every choice of translation term. They are primitive and have degree 32. 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 10. ALL THURSTON MULTIPLIERS c/d IN UNREDUCED FORM 0/16, 1/32, 2/8, 2/4, 2/2, 6/4, 2/1, 6/2, 10/2, 6/1, 14/2, 10/1, 14/1, 18/1 22/1, 26/1, 30/1, 31/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,0.033163) ( 0.033583,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 (0.032884,0.033264) 7/211 HST (0.033224,0.033288) 16/481 HST (0.033287,0.033291) 25/751 HST (0.033275,0.033306) 27/811 HST (0.033305,0.033307) 41/1231 HST (0.033299,0.033315) 43/1291 HST (0.033315,0.033315) 61/1831 HST (0.033311,0.033319) 62/1861 HST (0.033316,0.033322) 80/2401 HST (0.033320,0.033325) 102/3061 HST (0.033324,0.033325) 129/3871 HST (0.033323,0.033327) 135/4051 HST (0.033327,0.033339) 1/30 EXTENDED HST (0.033333,0.033370) 31/929 HST (0.033370,0.033370) 61/1828 HST (0.033370,0.033370) 698/20917 HST (0.033370,0.033370) 91/2727 EXTENDED HST (0.033370,0.033370) 1031/30896 HST (0.033370,0.033370) 121/3626 HST (0.033370,0.033371) 30/899 HST (0.033371,0.033371) 149/4465 HST (0.033371,0.033371) 89/2667 HST (0.033371,0.033371) 59/1768 HST (0.033371,0.033371) 147/4405 HST (0.033371,0.033371) 88/2637 HST (0.033371,0.033371) 293/8780 HST (0.033371,0.033371) 117/3506 HST (0.033371,0.033371) 146/4375 HST (0.033371,0.033372) 175/5244 HST (0.033372,0.033372) 1683/50432 HST (0.033372,0.033372) 29/869 EXTENDED HST (0.033371,0.033372) 434/13005 HST (0.033372,0.033372) 144/4315 HST (0.033372,0.033372) 115/3446 HST (0.033372,0.033372) 287/8600 HST (0.033372,0.033372) 86/2577 HST (0.033372,0.033372) 143/4285 HST (0.033372,0.033372) 57/1708 HST (0.033372,0.033373) 85/2547 HST (0.033373,0.033373) 28/839 HST (0.033373,0.033373) 111/3326 HST (0.033373,0.033374) 277/8300 HST (0.033374,0.033374) 83/2487 EXTENDED HST (0.033374,0.033374) 55/1648 HST (0.033374,0.033375) 27/809 HST (0.033375,0.033375) 53/1588 HST (0.033375,0.033377) 79/2367 HST (0.033376,0.033376) 26/779 HST (0.033377,0.033378) 76/2277 HST (0.033378,0.033378) 251/7520 HST (0.033378,0.033378) 326/9767 HST (0.033378,0.033378) 501/15010 HST (0.033378,0.033378) 25/749 EXTENDED HST (0.033377,0.033379) 74/2217 HST (0.033379,0.033379) 49/1468 HST (0.033379,0.033384) 23/689 HST (0.033383,0.033401) 20/599 HST (0.033391,0.033403) 18/539 HST (0.033398,0.033451) 14/419 HST (0.033413,0.033454) 12/359 HST (0.033433,0.033655) 8/239 HST 0.031286)(0.033489 5/149 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 -30/1 1 32 Yes Yes No No 0/1 1 32 Yes Yes No No -16750/559 1 32 Yes Yes No No -6202/207 1 32 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: 1971 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,0.033372) ( 0.033372,0.033372) ( 0.033372,0.033373) ( 0.033373,0.033373) ( 0.033373,0.033373) ( 0.033373,0.033374) ( 0.033374,0.033374) ( 0.033374,0.033374) ( 0.033374,0.033374) ( 0.033374,0.033375) ( 0.033375,0.033375) ( 0.033375,0.033387) ( 0.033387,0.033387) ( 0.033387,0.033387) ( 0.033387,0.033387) ( 0.033387,0.033387) ( 0.033387,0.033387) ( 0.033387,0.033387) ( 0.033387,0.033387) ( 0.033387,0.033405) ( 0.033405,0.033405) ( 0.033405,0.033405) ( 0.033405,0.033405) ( 0.033405,0.033405) ( 0.033405,0.033405) ( 0.033405,0.033405) ( 0.033407,0.033408) ( 0.033409,0.033417) ( 0.033419,0.033419) ( 0.033423,0.033430) ( 0.033434,0.033435) ( 0.033439,0.033452) ( 0.033456,0.033486) ( 0.033492,0.033494) ( 0.033500,0.033546) ( 0.033556,0.033671) ( 0.033704,0.033712) ( 0.033770,0.034154) ( 0.034425,0.034539) ( 0.034836,0.704545) ( 0.942962,1.064516) (33.507937,infinity) 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. 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,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>(2,32)(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)", "b=(1,32)(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,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>(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)", "d=<1,d,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>(1,2)(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: NewSphereMachine( "a=(1,32)(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=(1,32)(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,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>(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)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,32)(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,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>(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)", "c=(1,32)(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=(1,31)(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)", "a*b*c*d"); When the translation term of the affine map is lambda1+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=(1,32)(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)(31,32)", "a*b*c*d"); ****************************INTEGER OVERFLOW REPORT***************************** Imminent integer overflow halted evaluation of the slope function at slope -53838796/1796731 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -1445445/48238 during the search for all slope function fixed points. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 12885/386089. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 12885/386089. Imminent integer overflow halted evaluation of the slope function at slope -3810640/127173 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -6160674/205601 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -1561106/52099 during the search for all slope function fixed points. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 11711/350911. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 11711/350911. Imminent integer overflow halted evaluation of the slope function at slope -3141449/104840 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -1545622/51583 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -1373510/45839 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -1480840/49421 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -2021057/67450 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -2421463/80813 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -1928162/64351 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -3906801/130390 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -4172358/139253 during the search for all slope function fixed points. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 16127/483031. Imminent integer overflow halted evaluation of the slope function at slope -3462446/115601 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -16660541/556247 while computing a fixed point excluded interval about 355353/10643425. Imminent integer overflow halted evaluation of the slope function at slope -34938833/1166506 while computing a fixed point excluded interval about 355353/10643425. Imminent integer overflow halted evaluation of the slope function at slope -148071031/4943661 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -13651983/455800 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -8165641/272627 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -8223927/274573 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -2854157/95292 during the search for all slope function fixed points. Imminent integer overflow halted evaluation of the slope function at slope -6722089/224431 during the search for all slope function fixed points. Imminent integer overflow occurred during the computation of a fixed point excluded interval about 121155/3626851. Imminent integer overflow halted evaluation of the slope function at slope -12395655/414077 while computing a fixed point excluded interval about 121155/3626851.