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 only finitely many Thurston equivalence classes. However, this modular group Hurwitz class contains 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 0/2, 0/3, 0/6, 0/12, 0/24, 1/24, 1/8, 2/4, 2/3, 2/2, 4/3, 2/1, 4/1, 8/2 10/1, 12/1, 14/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,-1.165039) (-1.164829,-1.159073) (-1.158147,-1.149773) (-1.148444,-1.147882) (-1.147565,-1.145549) (-1.140014,-1.139282) (-1.135859,-1.134734) (-1.134162,-1.132331) (-1.129820,-1.127401) (-1.122110,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 (-1.165394,-1.164024) -106/91 HST (-1.162327,-1.155424) -73/63 HST (-1.150220,-1.149227) -238/207 HST (-1.149739,-1.147742) -54/47 HST (-1.148760,-1.146789) -101/88 HST (-1.145853,-1.144889) -63/55 HST (-1.144937,-1.144422) -87/76 HST (-1.144518,-1.144048) -111/97 HST (-1.144079,-1.141929) -8/7 EXTENDED HST (-1.141947,-1.141696) -153/134 HST (-1.141744,-1.141505) -121/106 HST (-1.141664,-1.141342) -234/205 HST (-1.141455,-1.141023) -97/85 HST (-1.141041,-1.140736) -81/71 HST (-1.140818,-1.140303) -73/64 HST (-1.140412,-1.140286) -65/57 HST (-1.140379,-1.139775) -122/107 HST (-1.139454,-1.138999) -90/79 HST (-1.139025,-1.138970) -295/259 HST (-1.138985,-1.138932) -418/367 HST (-1.138937,-1.138921) -746/655 HST (-1.138957,-1.138884) -992/871 HST (-1.138911,-1.138866) -41/36 EXTENDED HST (-1.138868,-1.138857) -1181/1037 HST (-1.138862,-1.138834) -812/713 HST (-1.138849,-1.138809) -566/497 HST (-1.138832,-1.138803) -484/425 HST (-1.138820,-1.138725) -238/209 HST (-1.138750,-1.138709) -197/173 HST (-1.138711,-1.138704) -1921/1687 HST (-1.138708,-1.138664) -156/137 EXTENDED HST (-1.138673,-1.138553) -115/101 HST (-1.138606,-1.138472) -263/231 HST (-1.138475,-1.138468) -1669/1466 HST (-1.138470,-1.138465) -2779/2441 HST (-1.138466,-1.138325) -74/65 HST (-1.138351,-1.138282) -502/441 HST (-1.138301,-1.138295) -107/94 EXTENDED HST (-1.138286,-1.138274) -675/593 HST (-1.138279,-1.138218) -247/217 HST (-1.138221,-1.138215) -1367/1201 HST (-1.138216,-1.138214) -2347/2062 HST (-1.138217,-1.138211) -3887/3415 HST (-1.138213,-1.138168) -140/123 HST (-1.138196,-1.138094) -173/152 HST (-1.138104,-1.138066) -511/449 HST (-1.138081,-1.138054) -272/239 HST (-1.138065,-1.138041) -643/565 HST (-1.138053,-1.138031) -338/297 HST (-1.138084,-1.137959) -404/355 HST (-1.137972,-1.137891) -33/29 EXTENDED HST (-1.137893,-1.137885) -949/834 HST (-1.137888,-1.137883) -883/776 HST (-1.137886,-1.137878) -817/718 HST (-1.137882,-1.137873) -751/660 HST (-1.137877,-1.137868) -685/602 HST (-1.137873,-1.137132) -91/80 HST (-1.137281,-1.137229) -58/51 EXTENDED HST (-1.137200,-1.136813) -83/73 HST (-1.136826,-1.136677) -133/117 HST (-1.136759,-1.136540) -183/161 HST (-1.136567,-1.136468) -308/271 HST (-1.136485,-1.136438) -508/447 HST (-1.136447,-1.136421) -708/623 HST (-1.136426,-1.136304) -25/22 EXTENDED HST (-1.136340,-1.136229) -642/565 HST (-1.136282,-1.136198) -417/367 HST (-1.136239,-1.136160) -317/279 HST (-1.136192,-1.136045) -242/213 HST (-1.136150,-1.136000) -192/169 HST (-1.136063,-1.135796) -117/103 HST (-1.135057,-1.134410) -219/193 HST (-1.134626,-1.134605) -59/52 EXTENDED HST (-1.134966,-1.133283) -76/67 HST (-1.133333,-1.129032) -43/38 EXTENDED HST -> HST (-1.130499,-1.130368) -26/23 EXTENDED HST (-1.128035,-1.124940) -62/55 HST (-1.125363,-1.124639) -9/8 EXTENDED HST (-1.125037,-1.120868) -73/65 HST (-1.123027,-1.121836) -55/49 HST The supplemental half-space computation shows that these NET maps are rational. SLOPE FUNCTION INFORMATION NUMBER OF FIXED POINTS: 3 EQUATOR? FIXED POINT c d 0 lambda1 lambda2 lambda1+lambda2 7/8 2 3 No No No No 8/9 1 8 No No No No 29/33 1 8 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: 6346 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,a*b,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,a*b,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*d,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,a*b,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");