These Thurston maps are NET maps for every choice of translation term. They are primitive and have degree 20. PURE MODULAR GROUP HURWITZ EQUIVALENCE CLASSES FOR TRANSLATIONS {0} {lambda1} {lambda2} {lambda1+lambda2} Since no Thurston multiplier is 1, this modular group Hurwitz class contains only finitely many Thurston equivalence classes. The number of pure modular group Hurwitz classes in this modular group Hurwitz class is 24. ALL THURSTON MULTIPLIERS c/d IN UNREDUCED FORM 0/1, 0/2, 0/4, 0/5, 0/10, 0/20, 1/20, 1/4, 2/5, 3/4, 2/1, 4/2, 4/1, 8/2 6/1, 8/1, 10/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,-0.133333) (-0.131715,-0.129179) (-0.121083,-0.118936) (-0.117647,-0.105263) (-0.104252,-0.102657) (-0.097477,-0.096081) (-0.094412,-0.087657) (-0.086265,-0.085170) (-0.081574,-0.080594) (-0.078269,-0.075622) (-0.073572,-0.072774) (-0.070132,-0.069407) (-0.067676,-0.065687) (-0.064135,-0.063528) (-0.061505,-0.060946) (-0.059608,-0.058060) (-0.053258,-0.052019) (-0.048132,-0.047117) (-0.043905,-0.043060) (-0.040361,-0.039645) (-0.037346,-0.036733) (-0.034751,-0.034219) (-0.032492,-0.032027) (-0.030510,-0.030099) (-0.028755,-0.028390) (-0.027191,-0.026865) (-0.025789,-0.025495) (-0.024524,-0.024258) (-0.023377,-0.023136) (-0.022333,-0.022112) (-0.021378,-0.021176) (-0.020502,-0.020315) ( 0.020315,0.020502 ) ( 0.021176,0.021378 ) ( 0.022112,0.022333 ) ( 0.023136,0.023377 ) ( 0.024258,0.024524 ) ( 0.025495,0.025789 ) ( 0.026865,0.027191 ) ( 0.028390,0.028755 ) ( 0.030099,0.030510 ) ( 0.032027,0.032492 ) ( 0.034219,0.034751 ) ( 0.036733,0.037346 ) ( 0.039645,0.040361 ) ( 0.043060,0.043905 ) ( 0.047117,0.048132 ) ( 0.052019,0.053258 ) ( 0.058060,0.059608 ) ( 0.060946,0.061505 ) ( 0.063528,0.064135 ) ( 0.065687,0.067676 ) ( 0.069407,0.070132 ) ( 0.072774,0.073572 ) ( 0.075622,0.078269 ) ( 0.080594,0.081574 ) ( 0.085170,0.086265 ) ( 0.087657,0.094412 ) ( 0.096081,0.097477 ) ( 0.102657,0.104252 ) ( 0.105263,0.117647 ) ( 0.118936,0.121083 ) ( 0.129179,0.131715 ) ( 0.133333,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 (-0.133532,-0.133135) -2/15 EXTENDED HST (-0.143693,-0.121427) -13/98 HST (-0.132635,-0.131521) -7/53 HST (-0.123176,-0.119575) -21/173 HST (-0.121377,-0.121048) -4/33 EXTENDED HST (-0.119095,-0.118196) -7/59 HST (-0.118577,-0.117647) -13/110 HST (-0.117802,-0.117493) -2/17 EXTENDED HST (-0.105387,-0.105139) -2/19 EXTENDED HST (-0.113522,-0.096039) -13/124 HST (-0.104827,-0.104130) -7/67 HST (-0.096185,-0.095598) -7/73 HST (-0.095847,-0.095238) -13/136 HST (-0.095340,-0.095137) -2/21 EXTENDED HST (-0.095238,-0.093458) -7/74 HST (-0.088496,-0.086957) -12/137 HST (-0.087569,-0.087189) -7/80 HST (-0.087041,-0.086872) -2/23 EXTENDED HST (-0.093822,-0.079432) -13/150 HST (-0.086659,-0.086182) -7/81 HST (-0.080000,-0.078740) -22/277 HST (-0.079416,-0.077471) -3/38 EXTENDED HST -> HST (-0.076383,-0.074582) -3/40 EXTENDED HST -> HST (-0.075188,-0.074074) -12/161 HST (-0.074518,-0.074243) -7/94 HST (-0.074135,-0.074013) -2/27 EXTENDED HST (-0.079948,-0.067721) -13/176 HST (-0.073858,-0.073511) -7/95 HST (-0.068531,-0.067078) -13/192 HST (-0.066261,-0.064901) -3/46 EXTENDED HST -> HST (-0.065359,-0.064516) -12/185 HST (-0.064853,-0.064644) -7/108 HST (-0.064563,-0.064470) -2/31 EXTENDED HST (-0.069649,-0.059020) -13/202 HST (-0.064352,-0.064089) -7/109 HST (-0.058507,-0.057444) -3/52 HST (-0.057803,-0.057143) -12/209 HST (-0.057407,-0.057243) -7/122 HST (-0.057179,-0.057106) -2/35 EXTENDED HST (-0.061701,-0.052300) -13/228 HST (-0.057014,-0.056807) -7/123 HST (-0.056844,-0.056366) -3/53 HST (-0.055590,-0.055521) -1/18 EXTENDED HST (-0.052378,-0.051525) -3/58 HST (-0.051813,-0.051282) -12/233 HST (-0.051494,-0.051363) -7/136 HST (-0.051311,-0.051253) -2/39 EXTENDED HST (-0.055381,-0.046954) -13/254 HST (-0.051178,-0.051012) -7/137 HST (-0.051041,-0.050655) -3/59 HST (-0.050028,-0.049972) -1/20 EXTENDED HST (-0.047412,-0.046711) -3/64 HST (-0.046948,-0.046512) -12/257 HST (-0.046686,-0.046578) -7/150 HST (-0.046536,-0.046487) -2/43 EXTENDED HST (-0.050235,-0.042599) -13/280 HST (-0.046426,-0.046289) -7/151 HST (-0.046313,-0.045996) -3/65 HST (-0.045478,-0.045431) -1/22 EXTENDED HST (-0.044390,-0.040801) -21/493 HST (-0.042573,-0.042533) -2/47 EXTENDED HST (-0.041686,-0.041647) -1/24 EXTENDED HST (-0.040816,-0.040486) -7/172 HST (-0.040664,-0.040148) -3/74 HST (-0.039854,-0.039358) -3/76 HST (-0.039526,-0.039216) -12/305 HST (-0.039340,-0.039263) -7/178 HST (-0.039233,-0.039198) -2/51 EXTENDED HST (-0.042363,-0.035934) -13/332 HST (-0.039155,-0.039057) -7/179 HST (-0.039075,-0.038848) -3/77 HST (-0.038478,-0.038445) -1/26 EXTENDED HST (-0.038060,-0.033720) -6/167 HST (-0.035729,-0.035700) -1/28 EXTENDED HST (-0.033793,-0.033623) -3/89 HST (-0.033626,-0.033601) -4/119 EXTENDED HST (-0.035519,-0.031454) -5/149 HST (-0.033346,-0.033321) -1/30 EXTENDED HST (-0.033295,-0.029473) -5/159 HST (-0.031261,-0.031239) -1/32 EXTENDED HST (-0.030226,-0.028699) -15/509 HST (-0.029421,-0.029402) -1/34 EXTENDED HST (-0.028497,-0.028242) -3/106 HST (-0.028329,-0.028169) -12/425 HST (-0.028233,-0.028193) -7/248 HST (-0.028178,-0.028160) -2/71 EXTENDED HST (-0.030438,-0.025831) -13/462 HST (-0.028138,-0.028087) -7/249 HST (-0.028096,-0.027979) -3/107 HST (-0.027786,-0.027769) -1/36 EXTENDED HST (-0.025912,-0.025702) -4/155 HST (-0.025581,-0.025375) -3/118 HST (-0.025445,-0.025316) -12/473 HST (-0.025368,-0.025336) -7/276 HST (-0.025324,-0.025309) -2/79 EXTENDED HST (-0.027358,-0.023219) -13/514 HST (-0.025291,-0.025250) -7/277 HST (-0.025258,-0.025163) -3/119 HST (-0.025007,-0.024993) -1/40 EXTENDED HST (-0.023206,-0.023037) -3/130 HST (-0.023095,-0.022989) -12/521 HST (-0.023031,-0.023005) -7/304 HST (-0.022994,-0.022983) -2/87 EXTENDED HST (-0.024843,-0.021087) -13/566 HST (-0.022968,-0.022934) -7/305 HST (-0.022940,-0.022862) -3/131 HST (-0.022733,-0.022721) -1/44 EXTENDED HST (-0.021088,-0.021066) -7/332 HST (-0.021072,-0.021060) -17/807 HST (-0.021062,-0.021056) -31/1472 HST (-0.021058,-0.021048) -2/95 EXTENDED HST (-0.022752,-0.019314) -13/618 HST (-0.021035,-0.021007) -7/333 HST (-0.021012,-0.020946) -3/143 HST (-0.020838,-0.020828) -1/48 EXTENDED HST (-0.065992,0.070665 ) 0/1 EXTENDED HST ( 0.065744,0.075738 ) 7/99 HST ( 0.071372,0.071486 ) 1/14 EXTENDED HST ( 0.077471,0.079416 ) 3/38 EXTENDED HST -> HST ( 0.078740,0.080000 ) 12/151 HST ( 0.079489,0.079804 ) 7/88 HST ( 0.079929,0.080072 ) 2/25 EXTENDED HST ( 0.073884,0.086679 ) 13/162 HST ( 0.080254,0.080667 ) 7/87 HST ( 0.086455,0.086957 ) 15/173 HST ( 0.086872,0.087041 ) 2/23 EXTENDED HST ( 0.086957,0.088496 ) 7/80 HST ( 0.093458,0.095238 ) 12/127 HST ( 0.094514,0.094960 ) 7/74 HST ( 0.095137,0.095340 ) 2/21 EXTENDED HST ( 0.088046,0.103229 ) 13/136 HST ( 0.095598,0.096185 ) 7/73 HST ( 0.104130,0.104827 ) 7/67 HST ( 0.104530,0.105263 ) 13/124 HST ( 0.105139,0.105387 ) 2/19 EXTENDED HST ( 0.117493,0.117802 ) 2/17 EXTENDED HST ( 0.108924,0.127590 ) 13/110 HST ( 0.118196,0.119095 ) 7/59 HST ( 0.118659,0.135957 ) 5/39 EXTENDED HST -> HST ( 0.128846,0.129219 ) 4/31 EXTENDED HST The supplemental half-space computation shows that these NET maps are rational. SLOPE FUNCTION INFORMATION There are no slope function fixed points. Number of excluded intervals computed by the fixed point finder: 3193 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=(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)", "b=(1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)", "c=(1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)", "d=<1,c^-1,c^-1,c^-1,c^-1,1,1,1,1,1,1,1,1,1,1,1,c,c,c,c>(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=(1,2)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)", "b=(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)", "c=<1,c^-1,c^-1,c^-1,c^-1,c^-1,1,1,1,1,1,1,1,1,1,c,c,c,c,c>(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)", "d=<1,d,c^-1,c^-1,c^-1,c^-1,1,1,1,1,1,1,1,1,1,1,c,c,c,c>(1,2)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)", "b=(1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)", "c=(1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)", "d=(1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)", "b=<1,c^-1,c^-1,c^-1,c^-1,c^-1,1,1,1,1,1,1,1,1,1,c,c,c,c,c>(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)", "c=(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)", "d=(1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)", "a*b*c*d");