These Thurston maps are NET maps for every choice of translation term. They are primitive and have degree 33. 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 12. ALL THURSTON MULTIPLIERS c/d IN UNREDUCED FORM 0/1, 0/3, 0/11, 0/33, 1/11, 1/3, 2/3, 1/1, 3/3, 2/1, 8/3, 4/1, 5/1, 7/1 8/1, 10/1, 12/1, 18/1, 30/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,-0.173395) (-0.160441,-0.155428) (-0.152296,-0.147772) (-0.138258,-0.134520) (-0.132167,-0.128747) (-0.121465,-0.118570) (-0.116738,-0.114062) (-0.108309,-0.106001) (-0.104535,-0.102384) (-0.097725,-0.095842) (-0.094642,-0.092875) (-0.089025,-0.087460) (-0.086459,-0.084982) (-0.081747,-0.080426) (-0.079579,-0.078326) (-0.075570,-0.074439) (-0.073713,-0.072637) (-0.070260,-0.069282) (-0.068652,-0.067718) (-0.065648,-0.064793) (-0.064242,-0.063423) (-0.061604,-0.060850) (-0.060364,-0.059640) ( 0.059640,0.060364 ) ( 0.060850,0.061604 ) ( 0.063423,0.064242 ) ( 0.064793,0.065648 ) ( 0.067718,0.068652 ) ( 0.069282,0.070260 ) ( 0.072637,0.073713 ) ( 0.074439,0.075570 ) ( 0.078326,0.079579 ) ( 0.080426,0.081747 ) ( 0.084982,0.086459 ) ( 0.087460,0.089025 ) ( 0.092875,0.094642 ) ( 0.095842,0.097725 ) ( 0.102384,0.104535 ) ( 0.106001,0.108309 ) ( 0.114062,0.116738 ) ( 0.118570,0.121465 ) ( 0.128747,0.132167 ) ( 0.134520,0.138258 ) ( 0.147772,0.152296 ) ( 0.155428,0.160441 ) ( 0.173395,0.181092 ) ( 0.182165,0.496148 ) ( 0.504546,0.984715 ) ( 1.015767,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 (-0.190476,-0.153846) -5/29 EXTENDED HST -> HST (-0.166921,-0.166413) -1/6 EXTENDED HST (-0.153954,-0.153738) -2/13 EXTENDED HST (-0.153754,-0.153475) -53/345 HST (-0.153623,-0.152948) -23/150 HST (-0.153326,-0.151767) -9/59 HST (-0.162162,-0.131579) -5/34 EXTENDED HST -> HST (-0.143044,-0.142671) -1/7 EXTENDED HST (-0.141176,-0.114943) -5/39 EXTENDED HST -> HST (-0.125143,-0.124857) -1/8 EXTENDED HST (-0.125000,-0.102041) -5/44 EXTENDED HST -> HST (-0.111224,-0.110998) -1/9 EXTENDED HST (-0.112150,-0.091743) -5/49 EXTENDED HST -> HST (-0.100091,-0.099909) -1/10 EXTENDED HST (-0.099145,-0.084289) -10/109 HST (-0.090985,-0.090834) -1/11 EXTENDED HST (-0.084350,-0.083876) -8/95 HST (-0.083930,-0.083772) -13/155 HST (-0.083940,-0.083658) -16/191 HST (-0.083766,-0.083565) -22/263 HST (-0.084455,-0.082655) -31/371 HST (-0.083397,-0.083270) -1/12 EXTENDED HST (-0.087540,-0.076127) -5/61 HST (-0.080903,-0.070385) -5/66 HST (-0.075201,-0.065447) -9/128 HST (-0.064889,-0.064608) -9/139 HST (-0.064675,-0.064555) -23/356 HST (-0.064598,-0.064510) -53/821 HST (-0.064535,-0.064497) -2/31 EXTENDED HST (-0.064500,-0.064451) -53/822 HST (-0.064477,-0.064358) -23/357 HST (-0.064424,-0.064148) -9/140 HST (-0.069364,-0.057143) -5/79 HST (-0.062536,-0.062464) -1/16 EXTENDED HST (-0.074484,0.064831 ) 0/1 EXTENDED HST ( 0.060241,0.071429 ) 5/76 HST ( 0.066626,0.066707 ) 1/15 EXTENDED HST ( 0.071382,0.071475 ) 1/14 EXTENDED HST ( 0.067883,0.077256 ) 5/69 HST ( 0.073209,0.083286 ) 5/64 HST ( 0.083270,0.083397 ) 1/12 EXTENDED HST ( 0.079443,0.090338 ) 5/59 HST ( 0.090198,0.090526 ) 15/166 HST ( 0.090400,0.090635 ) 22/243 HST ( 0.089897,0.091396 ) 31/342 HST ( 0.090834,0.090985 ) 1/11 EXTENDED HST ( 0.086837,0.098693 ) 5/54 HST ( 0.098574,0.099230 ) 8/81 HST ( 0.099155,0.099375 ) 13/131 HST ( 0.099140,0.099536 ) 16/161 HST ( 0.099384,0.099668 ) 22/221 HST ( 0.098871,0.100494 ) 31/311 HST ( 0.099909,0.100091 ) 1/10 EXTENDED HST ( 0.095748,0.108753 ) 5/49 EXTENDED HST -> HST ( 0.100000,0.117647 ) 6/55 HST ( 0.110998,0.111224 ) 1/9 EXTENDED HST ( 0.117584,0.117710 ) 2/17 EXTENDED HST ( 0.117701,0.117865 ) 53/450 HST ( 0.117778,0.118178 ) 23/195 HST ( 0.117953,0.118893 ) 9/76 HST ( 0.112360,0.131868 ) 5/41 EXTENDED HST -> HST ( 0.124857,0.125143 ) 1/8 EXTENDED HST ( 0.131769,0.132942 ) 9/68 HST ( 0.132658,0.133166 ) 23/173 HST ( 0.132986,0.133358 ) 53/398 HST ( 0.133252,0.133415 ) 2/15 EXTENDED HST ( 0.133403,0.133613 ) 53/397 HST ( 0.133502,0.134015 ) 23/172 HST ( 0.133727,0.134936 ) 9/67 HST ( 0.128205,0.150000 ) 5/36 EXTENDED HST -> HST ( 0.142671,0.143044 ) 1/7 EXTENDED HST ( 0.151767,0.153326 ) 9/59 HST ( 0.152948,0.153623 ) 23/150 HST ( 0.153384,0.153879 ) 53/345 HST ( 0.153738,0.153954 ) 2/13 EXTENDED HST ( 0.153938,0.154219 ) 53/344 HST ( 0.154070,0.154755 ) 23/149 HST ( 0.154370,0.155983 ) 9/58 HST ( 0.149254,0.173913 ) 5/31 EXTENDED HST -> HST ( 0.166413,0.166921 ) 1/6 EXTENDED HST ( 0.180565,0.181506 ) 23/127 HST ( 0.181173,0.181865 ) 53/292 HST ( 0.181667,0.181969 ) 2/11 EXTENDED HST ( 0.181947,0.182339 ) 49/269 HST ( 0.495579,0.496730 ) 65/131 HST ( 0.494109,0.499411 ) 76/153 HST ( 0.497726,0.502295 ) 1/2 EXTENDED HST ( 0.500848,0.508271 ) 56/111 HST ( 0.982472,0.987005 ) 65/66 HST ( 0.982943,0.990773 ) 76/77 HST ( 0.988411,0.993747 ) 108/109 HST ( 0.990946,1.009221 ) 1/1 EXTENDED HST ( 1.006697,1.012551 ) 108/107 HST ( 1.009401,1.017660 ) 76/75 HST ( 1.013342,1.018165 ) 65/64 HST The supplemental half-space computation shows that these NET maps are rational. SLOPE FUNCTION INFORMATION There are no slope function fixed points p/q with |p| <= 50 and |q| <= 50. Number of excluded intervals computed by the fixed point finder: 35515 The union of the excluded intervals computed by the fixed point finder became a union of 10,000 disjoint intervals: the search for all slope function fixed points aborted. 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 the nonslope. If the slope function maps slope p/q to slope p'/q', then |q'| <= |q| for every slope p/q with |p| <= 50 and |q| <= 50. 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,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1>(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=(1,33)(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)", "c=(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)", "d=(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: NewSphereMachine( "a=<1,1,b,b,b,b,b,b,b,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1>(1,2)(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,a*b,b,b,b,b,b,b,b,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1>(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=(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=(1,2)(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=(1,32)(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=(1,33)(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)", "c=(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)", "d=(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)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(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)", "b=(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,a*b,b,b,b,b,b,b,b,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1>(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)", "a*b*c*d"); ****************************INTEGER OVERFLOW REPORT***************************** Imminent integer overflow caused the modular group computation to abort.