These Thurston maps are NET maps for every choice of translation term. They are primitive and have degree 24. 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 0/4, 0/8, 0/24, 1/12, 1/8, 1/6, 1/4, 1/3, 1/2, 2/3, 1/1, 2/2, 3/3, 4/3, 2/1 3/1, 5/1, 9/1, 10/1, 12/1, 15/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,-1.154433) (-1.152080,-1.151016) (-1.147389,-1.146774) (-1.131438,-1.000000) (-1.000000,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 (-1.154478,-1.153197) -15/13 EXTENDED HST (-1.154532,-1.150370) -53/46 HST (-1.150943,-1.149660) -176/153 HST (-1.150030,-1.149970) -23/20 EXTENDED HST (-1.149715,-1.149501) -169/147 HST (-1.150130,-1.148727) -54/47 HST (-1.148945,-1.148064) -85/74 HST (-1.148168,-1.148128) -31/27 EXTENDED HST (-1.148438,-1.147059) -101/88 HST (-1.147701,-1.147391) -70/61 HST (-1.146787,-1.146756) -1297/1131 HST (-1.146772,-1.146567) -86/75 HST (-1.147028,-1.145937) -133/116 HST (-1.146354,-1.146329) -47/41 EXTENDED HST (-1.146129,-1.145031) -55/48 HST (-1.145129,-1.144824) -229/200 HST (-1.144931,-1.144924) -79/69 EXTENDED HST (-1.144855,-1.144801) -419/366 HST (-1.144847,-1.144753) -2609/2279 HST (-1.144800,-1.144678) -87/76 HST (-1.144680,-1.144674) -2785/2433 HST (-1.144677,-1.144632) -182/159 HST (-1.144840,-1.144370) -277/242 HST (-1.144581,-1.144575) -95/83 EXTENDED HST (-1.144452,-1.144255) -325/284 HST (-1.144341,-1.141500) -8/7 EXTENDED HST (-1.141536,-1.141349) -234/205 HST (-1.141416,-1.141413) -113/99 EXTENDED HST (-1.141590,-1.141135) -864/757 HST (-1.141344,-1.140608) -81/71 HST (-1.140628,-1.140622) -73/64 EXTENDED HST (-1.140647,-1.140538) -649/569 HST (-1.140596,-1.140571) -503/441 HST (-1.140587,-1.140276) -138/121 HST (-1.140355,-1.140346) -65/57 EXTENDED HST (-1.140777,-1.139831) -187/164 HST (-1.140236,-1.140136) -122/107 HST (-1.140135,-1.139851) -57/50 HST (-1.139850,-1.139717) -106/93 HST (-1.139979,-1.139401) -155/136 HST (-1.139546,-1.139524) -49/43 EXTENDED HST (-1.139685,-1.139115) -188/165 HST (-1.139375,-1.137718) -33/29 EXTENDED HST -> HST (-1.138253,-1.136947) -91/80 HST (-1.137468,-1.137024) -58/51 HST (-1.137283,-1.136490) -108/95 HST (-1.136508,-1.136444) -433/381 HST (-1.136464,-1.136420) -658/579 HST (-1.136433,-1.136402) -933/821 HST (-1.136413,-1.136391) -1358/1195 HST (-1.136448,-1.136333) -1908/1679 HST (-1.136388,-1.136339) -25/22 EXTENDED HST (-1.136547,-1.135572) -217/191 HST (-1.136109,-1.135887) -167/147 HST (-1.136004,-1.135629) -117/103 HST (-1.135838,-1.135586) -67/59 HST (-1.135802,-1.135135) -109/96 HST (-1.135296,-1.134979) -42/37 EXTENDED HST (-1.135082,-1.134442) -101/89 HST (-1.134806,-1.133895) -59/52 HST (-1.136331,-1.132272) -144/127 HST (-1.133398,-1.133267) -17/15 EXTENDED HST (-1.134615,-1.129630) -43/38 EXTENDED HST -> HST (-1.012677,-0.982419) -1/1 EXTENDED 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 7/8 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: 8823 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=(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)", "b=(1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)", "c=(1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)", "d=<1,a*b,a*b,a*b,a*b,c^-1,1,1,1,1,1,1,1,1,1,1,1,1,1,c*d,c*d,c*d,c*d,c*d>(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=(1,2)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)", "b=(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)", "c=<1,a*b,a*b,a*b,a*b,c^-1,c^-1,1,1,1,1,1,1,1,1,1,1,1,c,c*d,c*d,c*d,c*d,c*d>(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)", "d=<1,1,a*b,a*b,a*b,a*b,c^-1,1,1,1,1,1,1,1,1,1,1,1,1,c,c*d,c*d,c*d,c*d>(1,2)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)", "b=(1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)", "c=(1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)", "d=(1,23)(2,22)(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 lambda1+lambda2: NewSphereMachine( "a=(1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)", "b=<1,a*b,a*b,a*b,a*b,c^-1,c^-1,1,1,1,1,1,1,1,1,1,1,1,c,c*d,c*d,c*d,c*d,c*d>(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)", "c=(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)", "d=(1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)", "a*b*c*d");