These Thurston maps are NET maps for every choice of translation term. They are primitive and have degree 37. 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, 0/37, 1/1, 2/1, 3/1, 4/1, 5/1, 6/1, 7/1, 8/1, 9/1, 10/1, 11/1, 12/1 16/1, 22/1, 28/1, 34/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-1.484412,-0.500000) (-0.500000,0.500000 ) ( 0.500000,0.765568 ) ( 0.769876,0.851720 ) ( 0.852236,0.875505 ) ( 0.882944,0.899569 ) ( 0.913171,0.923953 ) ( 0.942857,0.948718 ) ( 0.971899,0.973968 ) ( 1.026032,1.028101 ) ( 1.051282,1.057143 ) ( 1.076047,1.086829 ) ( 1.100431,1.117056 ) ( 1.124495,1.147764 ) ( 1.148280,1.484412 ) The half-space computation does not determine rationality. EXCLUDED INTERVALS FOR JUST THE SUPPLEMENTAL HALF-SPACE COMPUTATION INTERVAL COMPUTED FOR HST OR EXTENDED HST (-17.687236,-16.311301) -17/1 EXTENDED HST -> HST (-16.647495,-15.348115) -16/1 EXTENDED HST -> HST (-15.607754,-14.384930) -15/1 EXTENDED HST -> HST (-14.568013,-13.421745) -14/1 EXTENDED HST -> HST (-13.528272,-12.458560) -13/1 EXTENDED HST -> HST (-12.488531,-11.495375) -12/1 EXTENDED HST -> HST (-12.022247,-10.947006) -126/11 HST (-11.448790,-10.532190) -11/1 EXTENDED HST -> HST (-11.370594,-9.499803 ) -21/2 EXTENDED HST -> HST (-10.409050,-9.569005 ) -10/1 EXTENDED HST -> HST (-10.039141,-8.894877 ) -47/5 EXTENDED HST -> HST ( -9.369309,-8.605819 ) -9/1 EXTENDED HST -> HST ( -9.173665,-7.664332 ) -17/2 EXTENDED HST -> HST ( -8.329568,-7.642634 ) -8/1 EXTENDED HST -> HST ( -8.075200,-6.746597 ) -15/2 EXTENDED HST -> HST ( -7.289827,-6.679449 ) -7/1 EXTENDED HST -> HST ( -6.976736,-5.828861 ) -13/2 EXTENDED HST -> HST ( -6.250086,-5.716264 ) -6/1 EXTENDED HST -> HST ( -5.878271,-4.911125 ) -11/2 EXTENDED HST -> HST ( -5.210345,-4.753079 ) -5/1 EXTENDED HST -> HST ( -4.779806,-3.993390 ) -9/2 EXTENDED HST -> HST ( -4.170604,-3.789894 ) -4/1 EXTENDED HST -> HST ( -4.091484,-3.511155 ) -15/4 EXTENDED HST -> HST ( -3.681341,-3.075654 ) -7/2 EXTENDED HST -> HST ( -3.130863,-2.826709 ) -3/1 EXTENDED HST -> HST ( -2.863790,-2.607659 ) -8/3 EXTENDED HST -> HST ( -2.732717,-2.421242 ) -13/5 EXTENDED HST -> HST ( -2.582877,-2.157919 ) -5/2 EXTENDED HST -> HST ( -2.428127,-1.888282 ) -15/7 EXTENDED HST -> HST ( -2.091122,-1.863523 ) -2/1 EXTENDED HST -> HST ( -2.130973,-1.594893 ) -11/6 EXTENDED HST -> HST ( -1.814679,-1.652378 ) -5/3 EXTENDED HST -> HST ( -1.668396,-1.478232 ) -11/7 EXTENDED HST -> HST ( -1.513641,-1.311616 ) -3/2 EXTENDED HST -> HST ( -0.502262,-0.497758 ) -1/2 EXTENDED HST ( 0.497758,0.502262 ) 1/2 EXTENDED HST ( 0.736454,0.815373 ) 10/13 EXTENDED HST -> HST ( 0.844442,0.861433 ) 23/27 EXTENDED HST -> HST ( 0.857268,0.910075 ) 15/17 EXTENDED HST -> HST ( 0.901316,0.923611 ) 21/23 EXTENDED HST -> HST ( 0.909574,0.939560 ) 25/27 EXTENDED HST -> HST ( 0.926446,0.929351 ) 13/14 EXTENDED HST -> HST ( 0.927279,0.954187 ) 16/17 EXTENDED HST -> HST ( 0.953634,0.956203 ) 21/22 EXTENDED HST -> HST ( 0.956104,0.956379 ) 131/137 HST ( 0.956228,0.957273 ) 22/23 EXTENDED HST -> HST ( 0.947846,0.967411 ) 23/24 EXTENDED HST -> HST ( 0.966949,0.968164 ) 30/31 EXTENDED HST -> HST ( 0.960234,0.976012 ) 61/63 HST ( 0.968730,0.968770 ) 31/32 EXTENDED HST ( 0.975814,0.976136 ) 163/167 HST ( 0.972064,0.980276 ) 450/461 HST ( 0.976179,0.976202 ) 41/42 EXTENDED HST ( 0.980235,0.980451 ) 50/51 HST ( 0.980421,0.980476 ) 301/307 HST ( 0.980456,0.980928 ) 51/52 HST ( 0.975777,0.986122 ) 103/105 HST ( 0.981127,0.981137 ) 52/53 EXTENDED HST ( 0.985401,0.986842 ) 72/73 HST ( 0.983721,0.990275 ) 75/76 HST ( 0.990260,0.990281 ) 611/617 HST ( 0.990273,0.990292 ) 917/926 HST ( 0.990290,0.990293 ) 102/103 EXTENDED HST ( 0.988996,0.991645 ) 205/207 HST ( 0.990383,0.990386 ) 103/104 EXTENDED HST ( 0.991635,0.991733 ) 119/120 HST ( 0.990899,0.992568 ) 2879/2903 HST ( 0.991734,0.991737 ) 120/121 EXTENDED HST ( 0.991005,0.994161 ) 401/404 HST ( 0.992584,0.992673 ) 134/135 HST ( 0.992779,0.995567 ) 171/172 HST ( 0.994806,0.996334 ) 225/226 HST ( 0.996332,0.996339 ) 272/273 HST ( 0.996338,0.996340 ) 1633/1639 HST ( 0.996339,0.996356 ) 273/274 HST ( 0.995353,0.997360 ) 547/549 HST ( 0.996363,0.996364 ) 274/275 EXTENDED HST ( 0.997254,1.002761 ) 1/1 EXTENDED HST ( 1.002426,1.003969 ) 314/313 HST ( 1.003202,1.003211 ) 313/312 HST ( 1.003215,1.003216 ) 312/311 EXTENDED HST ( 1.003224,1.003230 ) 311/310 HST ( 1.003235,1.003252 ) 310/309 HST ( 1.003236,1.004599 ) 257/256 HST ( 1.003917,1.005567 ) 212/211 HST ( 1.004750,1.006753 ) 175/174 HST ( 1.005779,1.009474 ) 132/131 HST ( 1.007663,1.007782 ) 130/129 HST ( 1.007767,1.012749 ) 99/98 HST ( 1.010266,1.010326 ) 98/97 HST ( 1.010327,1.010464 ) 97/96 HST ( 1.010525,1.010528 ) 96/95 EXTENDED HST ( 1.010617,1.015133 ) 78/77 HST ( 1.013108,1.018703 ) 64/63 HST ( 1.016004,1.016434 ) 62/61 HST ( 1.016555,1.016710 ) 61/60 HST ( 1.016945,1.016953 ) 60/59 EXTENDED HST ( 1.017127,1.024479 ) 49/48 EXTENDED HST -> HST ( 1.021073,1.029183 ) 40/39 EXTENDED HST -> HST ( 1.024700,1.035416 ) 34/33 EXTENDED HST -> HST ( 1.031230,1.031270 ) 33/32 EXTENDED HST ( 1.031930,1.053230 ) 25/24 EXTENDED HST -> HST ( 1.042727,1.043772 ) 24/23 EXTENDED HST -> HST ( 1.043797,1.046366 ) 23/22 EXTENDED HST -> HST ( 1.047587,1.047651 ) 22/21 EXTENDED HST ( 1.049955,1.050045 ) 21/20 EXTENDED HST ( 1.044521,1.074831 ) 18/17 EXTENDED HST -> HST ( 1.061404,1.064815 ) 17/16 EXTENDED HST -> HST ( 1.066607,1.066727 ) 16/15 EXTENDED HST ( 1.070649,1.073554 ) 15/14 EXTENDED HST -> HST ( 1.058511,1.093407 ) 43/40 EXTENDED HST -> HST ( 1.073511,1.125936 ) 11/10 EXTENDED HST -> HST ( 1.135834,1.158687 ) 31/27 EXTENDED HST -> HST ( 1.311616,1.513641 ) 3/2 EXTENDED HST -> HST ( 1.478232,1.668396 ) 8/5 EXTENDED HST -> HST ( 1.652378,1.814679 ) 5/3 EXTENDED HST -> HST ( 1.766111,1.924383 ) 11/6 EXTENDED HST -> HST ( 1.861862,1.984067 ) 27/14 EXTENDED HST -> HST ( 1.970928,1.996984 ) 125/63 HST ( 1.989217,2.067842 ) 2/1 EXTENDED HST -> HST ( 2.045757,2.241218 ) 15/7 EXTENDED HST -> HST ( 2.157919,2.582877 ) 5/2 EXTENDED HST -> HST ( 2.421242,2.732717 ) 13/5 EXTENDED HST -> HST ( 2.607659,2.863790 ) 8/3 EXTENDED HST -> HST ( 2.718030,3.039144 ) 23/8 EXTENDED HST -> HST ( 2.952402,3.107583 ) 3/1 EXTENDED HST -> HST ( 3.075654,3.681341 ) 7/2 EXTENDED HST -> HST ( 3.625136,3.962841 ) 15/4 EXTENDED HST -> HST ( 3.915587,4.147324 ) 4/1 EXTENDED HST -> HST ( 3.993390,4.779806 ) 9/2 EXTENDED HST -> HST ( 4.525000,5.147059 ) 24/5 EXTENDED HST -> HST ( 4.878773,5.187065 ) 5/1 EXTENDED HST -> HST ( 4.911125,5.878271 ) 11/2 EXTENDED HST -> HST ( 5.841958,6.226806 ) 6/1 EXTENDED HST -> HST ( 5.828861,6.976736 ) 13/2 EXTENDED HST -> HST ( 6.805143,7.266547 ) 7/1 EXTENDED HST -> HST ( 6.746597,8.075200 ) 15/2 EXTENDED HST -> HST ( 7.768328,8.306288 ) 8/1 EXTENDED HST -> HST ( 7.664332,9.173665 ) 17/2 EXTENDED HST -> HST ( 8.731513,9.346029 ) 9/1 EXTENDED HST -> HST ( 8.582068,10.272130 ) 19/2 EXTENDED HST -> HST ( 9.694698,10.385770 ) 10/1 EXTENDED HST -> HST ( 9.499803,11.370594 ) 21/2 EXTENDED HST -> HST ( 10.657884,11.425511 ) 11/1 EXTENDED HST -> HST ( 10.417539,12.469059 ) 23/2 EXTENDED HST -> HST ( 11.621069,12.465252 ) 12/1 EXTENDED HST -> HST ( 11.335274,13.567524 ) 25/2 EXTENDED HST -> HST ( 12.584254,13.504993 ) 13/1 EXTENDED HST -> HST ( 13.547439,14.544734 ) 14/1 EXTENDED HST -> HST ( 14.510624,15.584475 ) 15/1 EXTENDED HST -> HST ( 15.473809,16.624216 ) 16/1 EXTENDED HST -> HST ( 16.436994,17.663957 ) 17/1 EXTENDED HST -> HST ( 17.400180,18.703698 ) 18/1 EXTENDED HST -> HST ( 18.363365,19.743439 ) 19/1 EXTENDED HST -> HST -17.499023)(19.500000 infinity EXTENDED HST The supplemental half-space computation shows that these NET maps are rational. SLOPE FUNCTION INFORMATION There are no slope function fixed points because every nonzero multiplier is at least 1 and the map is rational. Similarly, there are not even any slope function cycles. 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 |p'| <= |p| for every slope p/q with |p| <= 50 and |q| <= 50. 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,d^-1,d*a,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)", "b=(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)", "c=<1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)", "d=(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)", "b=<1,a,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)", "c=<1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)", "d=(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)", "b=<1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)", "c=(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)", "d=<1,c^-1,b^-1*d^-1,c^-1,c,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)", "b=<1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)", "c=<1,a,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)", "d=(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)", "a*b*c*d"); ****************************INTEGER OVERFLOW REPORT***************************** Imminent integer overflow caused the modular group computation to abort.