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 (-infinity,-0.693967) ( 0.693967,0.870868 ) ( 0.871259,0.889288 ) ( 0.895210,0.908735 ) ( 0.920108,0.929328 ) ( 0.945946,0.951220 ) ( 0.972667,0.974628 ) ( 1.026728,1.028914 ) ( 1.054054,1.060606 ) ( 1.082306,1.095086 ) ( 1.111643,1.132575 ) ( 1.142198,1.173384 ) ( 1.174094,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 (-0.728864,-0.669852) -9/13 EXTENDED HST -> HST (-0.685185,-0.660714) -37/55 HST (-0.672727,-0.664671) -111/166 HST (-0.668675,-0.664671) -2/3 EXTENDED HST (-0.672727,-0.649123) -37/56 HST (-0.663511,-0.614250) -5/8 EXTENDED HST -> HST (-0.638870,-0.588438) -27/44 EXTENDED HST -> HST (-0.613047,-0.563631) -3/5 EXTENDED HST -> HST (-0.625000,-0.485714) -5/9 EXTENDED HST -> HST (-0.522033,-0.465696) -1/2 EXTENDED HST -> HST (-0.512821,-0.404762) -5/11 EXTENDED HST -> HST (-0.449279,-0.422154) -3/7 EXTENDED HST -> HST (-0.409059,-0.401235) -17/42 EXTENDED HST -> HST (-0.402174,-0.399281) -111/277 HST (-0.400722,-0.399281) -2/5 EXTENDED HST (-0.399716,-0.397966) -71/178 HST (-0.398862,-0.380517) -5/13 EXTENDED HST -> HST (-0.382901,-0.369705) -38/101 HST (-0.376238,-0.349515) -3/8 EXTENDED HST -> HST (-0.390540,-0.309976) -8/23 EXTENDED HST -> HST (-0.347370,-0.313768) -1/3 EXTENDED HST -> HST (-0.319328,-0.297521) -3/10 EXTENDED HST -> HST (-0.306587,-0.290614) -11/37 EXTENDED HST -> HST (-0.296560,-0.286298) -5/17 EXTENDED HST -> HST (-0.289062,-0.284615) -37/129 HST (-0.286822,-0.285347) -111/388 HST (-0.286082,-0.285347) -2/7 EXTENDED HST (-0.286822,-0.282443) -37/130 HST (-0.285133,-0.275634) -5/18 EXTENDED HST -> HST (-0.284449,-0.266545) -19/69 HST (-0.275362,-0.257143) -3/11 EXTENDED HST -> HST (-0.260284,-0.236585) -1/4 EXTENDED HST -> HST (-0.243590,-0.227848) -3/13 EXTENDED HST -> HST (-0.228728,-0.222575) -5/22 EXTENDED HST -> HST (-0.224242,-0.221557) -37/166 HST (-0.222892,-0.222000) -111/499 HST (-0.222445,-0.222000) -2/9 EXTENDED HST (-0.222892,-0.220238) -37/167 HST (-0.221871,-0.216076) -5/23 EXTENDED HST -> HST (-0.217143,-0.203390) -3/14 EXTENDED HST -> HST (-0.208110,-0.189878) -1/5 EXTENDED HST -> HST (-0.196891,-0.184615) -3/16 EXTENDED HST -> HST (-0.186150,-0.182054) -7/38 EXTENDED HST -> HST (-0.183168,-0.181373) -37/203 HST (-0.182266,-0.181669) -111/610 HST (-0.181967,-0.181669) -2/11 EXTENDED HST (-0.182266,-0.180488) -37/204 HST (-0.181583,-0.177683) -7/39 EXTENDED HST -> HST (-0.179245,-0.168224) -3/17 EXTENDED HST -> HST (-0.173360,-0.158572) -1/6 EXTENDED HST -> HST (-0.165217,-0.155172) -3/19 EXTENDED HST -> HST (-0.156937,-0.154015) -9/58 HST (-0.154167,-0.153740) -111/721 HST (-0.153953,-0.153740) -2/13 EXTENDED HST (-0.153995,-0.151806) -11/72 HST (-0.152610,-0.143426) -3/20 EXTENDED HST -> HST (-0.148555,-0.136128) -1/7 EXTENDED HST -> HST (-0.137709,-0.134391) -11/81 HST (-0.135648,-0.133460) -5/37 EXTENDED HST -> HST (-0.134058,-0.133094) -37/277 HST (-0.133574,-0.133253) -111/832 HST (-0.133413,-0.133253) -2/15 EXTENDED HST (-0.133574,-0.132616) -37/278 HST (-0.132867,-0.125000) -3/23 EXTENDED HST -> HST (-0.129960,-0.119250) -1/8 EXTENDED HST -> HST (-0.119446,-0.117746) -5/42 EXTENDED HST -> HST (-0.118211,-0.117460) -37/314 HST (-0.117834,-0.117585) -111/943 HST (-0.117709,-0.117585) -2/17 EXTENDED HST (-0.117834,-0.117089) -37/315 HST (-0.117548,-0.115902) -5/43 EXTENDED HST -> HST (-0.116836,-0.114439) -11/95 HST (-0.115502,-0.106095) -1/9 EXTENDED HST -> HST (-0.106701,-0.105342) -7/66 HST (-0.105714,-0.105114) -37/351 HST (-0.105413,-0.105213) -111/1054 HST (-0.105313,-0.105213) -2/19 EXTENDED HST (-0.105413,-0.104816) -37/352 HST (-0.105184,-0.103864) -5/48 EXTENDED HST -> HST (-0.103939,-0.095554) -1/10 EXTENDED HST -> HST (-0.095607,-0.095116) -37/388 HST (-0.095361,-0.095197) -111/1165 HST (-0.095279,-0.095197) -2/21 EXTENDED HST (-0.095361,-0.094872) -37/389 HST (-0.095173,-0.094091) -7/74 HST (-0.094480,-0.086919) -1/11 EXTENDED HST -> HST (-0.087004,-0.086301) -11/127 HST (-0.086599,-0.079715) -1/12 EXTENDED HST -> HST (-0.079932,-0.073613) -1/13 EXTENDED HST -> HST (-0.074218,-0.068380) -1/14 EXTENDED HST -> HST (-0.069266,-0.063841) -1/15 EXTENDED HST -> HST (-0.064934,-0.059867) -1/16 EXTENDED HST -> HST (-0.061112,-0.056359) -1/17 EXTENDED HST -> HST (-0.057143,0.051285 ) 0/1 EXTENDED HST ( 0.050794,0.054611 ) 1/19 EXTENDED HST -> HST ( 0.053627,0.057643 ) 1/18 EXTENDED HST -> HST ( 0.056793,0.061032 ) 1/17 EXTENDED HST -> HST ( 0.060357,0.064844 ) 1/16 EXTENDED HST -> HST ( 0.064399,0.069163 ) 1/15 EXTENDED HST -> HST ( 0.069020,0.074099 ) 1/14 EXTENDED HST -> HST ( 0.074040,0.074556 ) 11/148 HST ( 0.074356,0.079795 ) 1/13 EXTENDED HST -> HST ( 0.079647,0.079951 ) 17/213 HST ( 0.079909,0.079995 ) 65/813 HST ( 0.079971,0.080029 ) 2/25 EXTENDED HST ( 0.080011,0.080082 ) 71/887 HST ( 0.080046,0.080828 ) 7/87 HST ( 0.080586,0.086438 ) 1/12 EXTENDED HST -> HST ( 0.085999,0.086903 ) 9/104 HST ( 0.086860,0.086943 ) 71/817 HST ( 0.086922,0.086991 ) 2/23 EXTENDED HST ( 0.086970,0.087053 ) 71/816 HST ( 0.087010,0.087935 ) 5/57 HST ( 0.085907,0.089995 ) 27/307 HST ( 0.087956,0.094288 ) 1/11 EXTENDED HST -> HST ( 0.094091,0.095173 ) 5/53 HST ( 0.095122,0.095222 ) 71/746 HST ( 0.095197,0.095279 ) 2/21 EXTENDED HST ( 0.095238,0.100000 ) 3/31 EXTENDED HST -> HST ( 0.096809,0.103706 ) 1/10 EXTENDED HST -> HST ( 0.102687,0.104613 ) 11/106 HST ( 0.103864,0.105184 ) 5/48 EXTENDED HST -> HST ( 0.105122,0.105243 ) 71/675 HST ( 0.105213,0.105313 ) 2/19 EXTENDED HST ( 0.105290,0.105738 ) 19/180 HST ( 0.105572,0.110787 ) 3/28 EXTENDED HST -> HST ( 0.107645,0.115215 ) 1/9 EXTENDED HST -> HST ( 0.111455,0.116923 ) 3/26 EXTENDED HST -> HST ( 0.116450,0.117734 ) 11/94 HST ( 0.117585,0.117709 ) 2/17 EXTENDED HST ( 0.117672,0.117824 ) 71/603 HST ( 0.117746,0.119446 ) 9/76 HST ( 0.118421,0.124183 ) 3/25 EXTENDED HST -> HST ( 0.121211,0.129596 ) 1/8 EXTENDED HST -> HST ( 0.125874,0.131944 ) 3/23 EXTENDED HST -> HST ( 0.131096,0.133207 ) 7/53 HST ( 0.133107,0.133302 ) 71/533 HST ( 0.133253,0.133413 ) 2/15 EXTENDED HST ( 0.133365,0.133561 ) 71/532 HST ( 0.133460,0.135648 ) 7/52 HST ( 0.134831,0.141264 ) 3/22 EXTENDED HST -> HST ( 0.138690,0.148081 ) 1/7 EXTENDED HST -> HST ( 0.144578,0.151394 ) 3/20 EXTENDED HST -> HST ( 0.150875,0.153678 ) 5/33 EXTENDED HST -> HST ( 0.153544,0.153804 ) 71/462 HST ( 0.153740,0.153953 ) 2/13 EXTENDED HST ( 0.153888,0.154149 ) 71/461 HST ( 0.154015,0.156937 ) 5/32 EXTENDED HST -> HST ( 0.156522,0.163793 ) 3/19 EXTENDED HST -> HST ( 0.162059,0.172715 ) 1/6 EXTENDED HST -> HST ( 0.169811,0.177570 ) 3/17 EXTENDED HST -> HST ( 0.173861,0.181305 ) 19/107 HST ( 0.177683,0.181583 ) 5/28 EXTENDED HST -> HST ( 0.181397,0.181759 ) 71/391 HST ( 0.181669,0.181967 ) 2/11 EXTENDED HST ( 0.181877,0.182242 ) 71/390 HST ( 0.182054,0.186150 ) 5/27 EXTENDED HST -> HST ( 0.183790,0.190052 ) 11/59 HST ( 0.186528,0.194872 ) 3/16 EXTENDED HST -> HST ( 0.181108,0.207920 ) 38/195 HST ( 0.194899,0.207181 ) 1/5 EXTENDED HST -> HST ( 0.205714,0.214689 ) 3/14 EXTENDED HST -> HST ( 0.212546,0.216842 ) 38/177 HST ( 0.216076,0.221871 ) 5/23 EXTENDED HST -> HST ( 0.221593,0.222135 ) 71/320 HST ( 0.222000,0.222445 ) 2/9 EXTENDED HST ( 0.222310,0.222855 ) 71/319 HST ( 0.222575,0.228728 ) 5/22 EXTENDED HST -> HST ( 0.224342,0.233337 ) 27/118 HST ( 0.228896,0.236642 ) 3/13 EXTENDED HST -> HST ( 0.223684,0.253165 ) 5/21 EXTENDED HST -> HST ( 0.244432,0.258831 ) 1/4 EXTENDED HST -> HST ( 0.246377,0.277778 ) 5/19 EXTENDED HST -> HST ( 0.264956,0.275391 ) 3/11 EXTENDED HST -> HST ( 0.275634,0.285133 ) 5/18 EXTENDED HST -> HST ( 0.284675,0.285569 ) 71/249 HST ( 0.285347,0.286082 ) 2/7 EXTENDED HST ( 0.285859,0.286761 ) 71/248 HST ( 0.286298,0.296560 ) 5/17 EXTENDED HST -> HST ( 0.291433,0.302070 ) 27/91 HST ( 0.296841,0.310002 ) 3/10 EXTENDED HST -> HST ( 0.295652,0.330579 ) 5/16 EXTENDED HST -> HST ( 0.327720,0.344787 ) 1/3 EXTENDED HST -> HST ( 0.336634,0.373832 ) 5/14 EXTENDED HST -> HST ( 0.360463,0.380055 ) 3/8 EXTENDED HST -> HST ( 0.374074,0.386084 ) 27/71 HST ( 0.380517,0.398862 ) 5/13 EXTENDED HST -> HST ( 0.397966,0.399716 ) 71/178 HST ( 0.399281,0.400722 ) 2/5 EXTENDED HST ( 0.400284,0.402055 ) 71/177 HST ( 0.401145,0.421585 ) 5/12 EXTENDED HST -> HST ( 0.415770,0.428203 ) 27/64 HST ( 0.422154,0.449279 ) 3/7 EXTENDED HST -> HST ( 0.435897,0.476190 ) 5/11 EXTENDED HST -> HST ( 0.454581,0.492753 ) 10/21 EXTENDED HST -> HST ( 0.483596,0.502710 ) 34/69 HST ( 0.497107,0.516221 ) 1/2 EXTENDED HST -> HST ( 0.508721,0.546880 ) 9/17 EXTENDED HST -> HST ( 0.531250,0.571429 ) 5/9 EXTENDED HST -> HST ( 0.563631,0.613047 ) 3/5 EXTENDED HST -> HST ( 0.607120,0.619210 ) 27/44 EXTENDED HST -> HST ( 0.614250,0.663511 ) 5/8 EXTENDED HST -> HST ( 0.661036,0.665878 ) 71/107 HST ( 0.664671,0.668675 ) 2/3 EXTENDED HST ( 0.667457,0.672394 ) 71/106 HST ( 0.669852,0.728864 ) 9/13 EXTENDED HST -> HST ( 0.864224,0.879363 ) 27/31 EXTENDED HST -> HST ( 0.871842,0.919762 ) 17/19 EXTENDED HST -> HST ( 0.909091,0.929936 ) 23/25 EXTENDED HST -> HST ( 0.915842,0.943878 ) 27/29 EXTENDED HST -> HST ( 0.931486,0.934013 ) 14/15 EXTENDED HST -> HST ( 0.931306,0.956793 ) 17/18 EXTENDED HST -> HST ( 0.955689,0.958040 ) 45/47 EXTENDED HST -> HST ( 0.957949,0.958203 ) 137/143 HST ( 0.958063,0.959024 ) 23/24 EXTENDED HST -> HST ( 0.949951,0.968752 ) 24/25 EXTENDED HST -> HST ( 0.961455,0.976761 ) 63/65 HST ( 0.969678,0.969715 ) 32/33 EXTENDED HST ( 0.971440,0.982852 ) 85/87 HST ( 0.977059,0.977651 ) 43/44 EXTENDED HST -> HST ( 0.981405,0.984323 ) 115/117 HST ( 0.982968,0.983213 ) 58/59 HST ( 0.981690,0.987026 ) 63/64 HST ( 0.985125,0.989471 ) 77/78 HST ( 0.988781,0.990163 ) 4229/4274 HST ( 0.989472,0.989476 ) 94/95 EXTENDED HST ( 0.989629,0.990710 ) 302/305 HST ( 0.990194,0.990199 ) 101/102 EXTENDED HST ( 0.989184,0.992353 ) 107/108 HST ( 0.992278,0.992395 ) 130/131 HST ( 0.990632,0.994244 ) 131/132 HST ( 0.993541,1.006543 ) 1/1 EXTENDED HST ( 1.005823,1.009547 ) 131/130 HST ( 1.007722,1.007843 ) 129/128 HST ( 1.007847,1.012881 ) 98/97 HST ( 1.010373,1.010433 ) 97/96 HST ( 1.010435,1.010574 ) 96/95 HST ( 1.010637,1.010640 ) 95/94 EXTENDED HST ( 1.010751,1.010755 ) 94/93 EXTENDED HST ( 1.010755,1.015331 ) 77/76 HST ( 1.013320,1.019006 ) 63/62 HST ( 1.016265,1.016708 ) 61/60 HST ( 1.016834,1.016994 ) 60/59 HST ( 1.017237,1.017245 ) 59/58 EXTENDED HST ( 1.017490,1.025001 ) 48/47 EXTENDED HST -> HST ( 1.021611,1.029940 ) 39/38 EXTENDED HST -> HST ( 1.025461,1.036518 ) 33/32 EXTENDED HST -> HST ( 1.032237,1.032279 ) 32/31 EXTENDED HST ( 1.033331,1.055617 ) 24/23 EXTENDED HST -> HST ( 1.044634,1.045776 ) 23/22 EXTENDED HST -> HST ( 1.045803,1.048620 ) 22/21 EXTENDED HST -> HST ( 1.049964,1.050036 ) 21/20 EXTENDED HST ( 1.052582,1.052682 ) 20/19 EXTENDED HST ( 1.047294,1.079635 ) 17/16 EXTENDED HST -> HST ( 1.065421,1.069307 ) 16/15 EXTENDED HST -> HST ( 1.071360,1.071498 ) 15/14 EXTENDED HST ( 1.076020,1.079393 ) 14/13 EXTENDED HST -> HST ( 1.063218,1.101190 ) 27/25 EXTENDED HST -> HST ( 1.081389,1.140166 ) 10/9 EXTENDED HST -> HST ( 1.139064,1.141784 ) 65/57 HST ( 1.121212,1.163522 ) 153/134 HST ( 1.159000,1.186390 ) 27/23 EXTENDED HST -> 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,a*b,b,b,b,b,b,b,b,b,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1>(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=(1,37)(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)", "c=(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)", "d=(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: NewSphereMachine( "a=<1,1,b,b,b,b,b,b,b,b,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1>(1,2)(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,a*b,b,b,b,b,b,b,b,b,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1>(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=(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=(1,2)(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)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)", "b=(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)", "c=(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)", "d=(1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(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)", "b=(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,a*b,b,b,b,b,b,b,b,b,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1>(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)", "a*b*c*d"); ****************************INTEGER OVERFLOW REPORT***************************** Imminent integer overflow caused the modular group computation to abort.