These Thurston maps are NET maps for every choice of translation term. They are primitive and have degree 21. 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 24. ALL THURSTON MULTIPLIERS c/d IN UNREDUCED FORM 0/7, 0/21, 1/21, 1/7, 1/3, 2/3, 1/1, 2/1, 3/1, 4/1, 5/1, 7/1, 8/1, 9/1, 13/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,-1.189599) (-1.186061,-1.176193) (-1.172292,-1.170509) (-1.162701,-1.161732) (-1.161657,-1.159013) (-1.148712,-1.145442) (-1.141434,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 (-1.189655,-1.187500) -44/37 EXTENDED HST -> HST (-1.187642,-1.187360) -19/16 EXTENDED HST (-1.188283,-1.184634) -70/59 HST (-1.177215,-1.173913) -167/142 HST (-1.176002,-1.175310) -87/74 HST (-1.175502,-1.175367) -67/57 HST (-1.175023,-1.174978) -47/40 EXTENDED HST (-1.174001,-1.173824) -27/23 EXTENDED HST (-1.173856,-1.173706) -412/351 HST (-1.173787,-1.173688) -304/259 HST (-1.173739,-1.173275) -142/121 HST (-1.173548,-1.173278) -88/75 HST (-1.173286,-1.173259) -1036/883 HST (-1.173271,-1.173190) -149/127 HST (-1.173201,-1.172469) -61/52 HST (-1.172482,-1.171184) -34/29 EXTENDED HST -> HST (-1.170671,-1.169677) -55/47 HST (-1.169797,-1.169589) -131/112 HST (-1.169606,-1.169507) -269/230 HST (-1.169523,-1.169460) -69/59 EXTENDED HST (-1.169543,-1.169308) -352/301 HST (-1.169434,-1.169390) -283/242 HST (-1.169419,-1.169341) -214/183 HST (-1.169384,-1.168603) -76/65 HST (-1.168940,-1.168366) -104/89 HST (-1.168458,-1.167961) -118/101 HST (-1.168247,-1.165272) -7/6 EXTENDED HST (-1.165310,-1.165178) -275/236 HST (-1.165223,-1.164710) -106/91 HST (-1.164725,-1.164687) -99/85 EXTENDED HST (-1.164699,-1.164121) -85/73 HST (-1.164375,-1.163550) -64/55 HST (-1.163722,-1.163325) -185/159 HST (-1.163482,-1.163369) -121/104 HST (-1.163621,-1.163044) -406/349 HST (-1.163298,-1.163233) -57/49 EXTENDED HST (-1.163066,-1.161999) -50/43 EXTENDED HST -> HST (-1.161910,-1.161331) -194/167 HST (-1.159125,-1.158354) -73/63 HST (-1.159930,-1.156811) -139/120 HST (-1.157953,-1.157837) -22/19 EXTENDED HST (-1.157766,-1.155920) -37/32 EXTENDED HST -> HST (-1.155949,-1.155727) -89/77 HST (-1.155817,-1.155664) -334/289 HST (-1.155708,-1.154946) -52/45 HST (-1.154958,-1.154722) -82/71 HST (-1.155952,-1.153348) -112/97 HST (-1.153939,-1.153755) -15/13 EXTENDED HST (-1.154425,-1.151486) -83/72 HST (-1.152613,-1.149022) -38/33 EXTENDED HST -> HST (-1.150090,-1.149910) -23/20 EXTENDED HST (-1.149752,-1.148037) -54/47 HST (-1.145785,-1.144043) -71/62 HST (-1.144485,-1.143507) -143/125 HST (-1.143671,-1.143305) -255/223 HST (-1.143307,-1.143301) -1460/1277 HST (-1.143304,-1.143300) -367/321 HST (-1.143458,-1.143128) -375/328 HST (-1.143225,-1.142491) -8/7 EXTENDED HST (-1.142628,-1.142271) -401/351 HST (-1.142449,-1.141956) -249/218 HST (-1.142192,-1.141392) -153/134 HST (-1.141739,-1.141181) -121/106 HST The supplemental half-space computation shows that these NET maps are rational. SLOPE FUNCTION INFORMATION NUMBER OF FIXED POINTS: 1 EQUATOR? FIXED POINT c d 0 lambda1 lambda2 lambda1+lambda2 6/7 1 3 No No No No NUMBER OF EQUATORS: 0 0 0 0 There are no more slope function fixed points. Number of excluded intervals computed by the fixed point finder: 5135 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,c^-1*b,c^-1*b,c^-1*b,c^-1*b,b,b,b,b,1,1,1,b^-1,b^-1,b^-1,b^-1*c,b^-1*c,b^-1*c,b^-1*c,b^-1*c>(2,21)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)", "b=(1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)", "c=(1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)", "d=<1,a*b,a*b,a*b,a*b,c^-1,1,1,1,1,1,1,1,1,1,1,c*d,c*d,c*d,c*d,c*d>(2,21)(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 lambda1: NewSphereMachine( "a=(1,2)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)", "b=<1,b,c^-1*b,c^-1*b,c^-1*b,c^-1*b,b,b,b,b,1,1,1,b^-1,b^-1,b^-1,b^-1*c,b^-1*c,b^-1*c,b^-1*c,b^-1*c>(2,21)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)", "c=<1,a*b,a*b,a*b,a*b,c^-1,1,1,1,1,1,1,1,1,1,1,c*d,c*d,c*d,c*d,c*d>(2,21)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)", "d=<1,1,a*b,a*b,a*b,a*b,1,1,1,1,1,1,1,1,1,1,1,c*d,c*d,c*d,c*d>(1,2)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)", "b=(1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)", "c=(1,21)(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"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)", "b=<1,a*b,a*b,a*b,a*b,c^-1,1,1,1,1,1,1,1,1,1,1,c*d,c*d,c*d,c*d,c*d>(2,21)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)", "c=<1,b,c^-1*b,c^-1*b,c^-1*b,c^-1*b,b,b,b,b,1,1,1,b^-1,b^-1,b^-1,b^-1*c,b^-1*c,b^-1*c,b^-1*c,b^-1*c>(2,21)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)", "d=(1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)", "a*b*c*d");