These Thurston maps are NET maps for every choice of translation term. They have degree 40. They are imprimitive, each factoring as a NET map with degree 20 followed by a Euclidean NET map with degree 2. 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 0/4, 0/20, 1/10, 1/4, 1/2, 2/4, 2/2, 4/4, 3/2, 4/2, 5/2, 7/2 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,-1.005892) (-0.992731,-0.201178) (-0.196639,-0.088269) (-0.088169,-0.085777) (-0.085683,-0.084538) (-0.082163,-0.081109) (-0.081025,-0.079001) (-0.074952,-0.073217) (-0.069726,-0.068222) (-0.065181,-0.063865) (-0.061192,-0.060031) (-0.057664,-0.056631) (-0.054520,-0.053596) (-0.051701,-0.050870) (-0.049160,-0.048407) (-0.046856,-0.046172) (-0.044759,-0.044134) (-0.042841,-0.042269) (-0.041081,-0.040555) ( 0.040555,0.041081 ) ( 0.042269,0.042841 ) ( 0.044134,0.044759 ) ( 0.046172,0.046856 ) ( 0.048407,0.049160 ) ( 0.050870,0.051701 ) ( 0.053596,0.054520 ) ( 0.056631,0.057664 ) ( 0.060031,0.061192 ) ( 0.063865,0.065181 ) ( 0.068222,0.069726 ) ( 0.073217,0.074952 ) ( 0.079001,0.081025 ) ( 0.081109,0.082163 ) ( 0.084538,0.085683 ) ( 0.085777,0.088169 ) ( 0.088269,0.089518 ) ( 0.092344,0.093712 ) ( 0.093825,0.140845 ) ( 0.144928,0.330110 ) ( 0.336620,0.992731 ) ( 1.007376,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 (-1.016065,-0.984434) -1/1 EXTENDED HST (-0.201729,-0.200573) -35/174 HST (-0.200634,-0.199370) -1/5 EXTENDED HST (-0.199513,-0.199064) -56/281 HST (-0.199283,-0.198624) -39/196 HST (-0.198954,-0.197995) -27/136 HST (-0.198492,-0.197117) -18/91 HST (-0.197746,-0.195707) -12/61 HST (-0.095861,-0.081825) -3/34 EXTENDED HST -> HST (-0.084615,-0.076271) -3/37 EXTENDED HST -> HST (-0.076421,-0.075962) -9/118 HST (-0.076255,-0.075647) -6/79 HST (-0.075925,-0.075024) -4/53 HST (-0.075423,-0.073847) -3/40 EXTENDED HST -> HST (-0.077465,-0.069231) -3/41 EXTENDED HST -> HST (-0.071509,-0.071348) -1/14 EXTENDED HST (-0.072288,-0.062171) -3/44 EXTENDED HST -> HST (-0.066737,-0.066596) -1/15 EXTENDED HST (-0.063953,-0.057692) -3/49 EXTENDED HST -> HST (-0.057942,-0.057008) -3/52 HST (-0.059783,-0.053571) -3/53 HST (-0.055604,-0.055507) -1/18 EXTENDED HST (-0.056864,-0.048985) -3/56 HST (-0.052675,-0.052588) -1/19 EXTENDED HST (-0.048879,-0.048213) -3/62 HST (-0.048377,-0.048010) -4/83 HST (-0.048122,-0.047879) -6/125 HST (-0.048009,-0.047686) -9/188 HST (-0.047716,-0.047651) -35/734 HST (-0.047655,-0.047583) -1/21 EXTENDED HST (-0.047591,-0.047566) -56/1177 HST (-0.047578,-0.047541) -39/820 HST (-0.047559,-0.047505) -27/568 HST (-0.047533,-0.047454) -18/379 HST (-0.047490,-0.047372) -12/253 HST (-0.047426,-0.047249) -9/190 HST (-0.047362,-0.047127) -6/127 HST (-0.047235,-0.046884) -4/85 HST (-0.047040,-0.046422) -3/64 HST (-0.048673,-0.043689) -3/65 HST (-0.045487,-0.045422) -1/22 EXTENDED HST (-0.045833,-0.041284) -9/206 HST (-0.043508,-0.043448) -1/23 EXTENDED HST (-0.042969,-0.038793) -3/73 HST (-0.052629,0.052637 ) 0/1 EXTENDED HST ( 0.052635,0.052639 ) 512/9727 HST ( 0.052638,0.052671 ) 120/2279 HST ( 0.052655,0.052676 ) 82/1557 HST ( 0.052665,0.052697 ) 56/1063 HST ( 0.052681,0.052728 ) 39/740 HST ( 0.052705,0.052772 ) 27/512 HST ( 0.052737,0.052835 ) 18/341 HST ( 0.052790,0.052937 ) 12/227 HST ( 0.052869,0.053091 ) 9/170 HST ( 0.052949,0.053246 ) 6/113 HST ( 0.053109,0.053559 ) 4/75 HST ( 0.053358,0.054175 ) 3/56 HST ( 0.051724,0.056701 ) 3/55 HST ( 0.055507,0.055604 ) 1/18 EXTENDED HST ( 0.053906,0.061789 ) 3/52 HST ( 0.058769,0.058878 ) 1/17 EXTENDED HST ( 0.061655,0.062058 ) 6/97 HST ( 0.057906,0.065702 ) 9/145 HST ( 0.062438,0.062562 ) 1/16 EXTENDED HST ( 0.065359,0.066225 ) 5/76 HST ( 0.065916,0.066537 ) 10/151 HST ( 0.066477,0.066603 ) 35/526 HST ( 0.066596,0.066737 ) 1/15 EXTENDED HST ( 0.066721,0.066771 ) 56/839 HST ( 0.066747,0.066821 ) 39/584 HST ( 0.066784,0.066892 ) 27/404 HST ( 0.066836,0.066993 ) 18/269 HST ( 0.066921,0.067158 ) 12/179 HST ( 0.067049,0.067406 ) 9/134 HST ( 0.067177,0.067656 ) 6/89 HST ( 0.067435,0.068162 ) 4/59 HST ( 0.067836,0.069163 ) 3/44 EXTENDED HST -> HST ( 0.066176,0.072368 ) 3/43 EXTENDED HST -> HST ( 0.071348,0.071509 ) 1/14 EXTENDED HST ( 0.072312,0.073148 ) 4/55 HST ( 0.072993,0.074074 ) 3/41 EXTENDED HST -> HST ( 0.070317,0.080439 ) 3/40 EXTENDED HST -> HST ( 0.076830,0.077017 ) 1/13 EXTENDED HST ( 0.080000,0.081301 ) 3/37 EXTENDED HST -> HST ( 0.081837,0.082549 ) 6/73 HST ( 0.077279,0.087198 ) 9/109 HST ( 0.083224,0.083443 ) 1/12 EXTENDED HST ( 0.086644,0.088821 ) 3/34 EXTENDED HST -> HST ( 0.089132,0.089977 ) 6/67 HST ( 0.089520,0.090668 ) 9/100 HST ( 0.090556,0.090791 ) 35/386 HST ( 0.090779,0.091040 ) 1/11 EXTENDED HST ( 0.091010,0.091104 ) 56/615 HST ( 0.091058,0.091196 ) 39/428 HST ( 0.091127,0.091329 ) 27/296 HST ( 0.091224,0.091517 ) 18/197 HST ( 0.091382,0.091825 ) 12/131 HST ( 0.091621,0.092289 ) 9/98 HST ( 0.091861,0.092759 ) 6/65 HST ( 0.093098,0.095615 ) 3/32 EXTENDED HST -> HST ( 0.139456,0.142262 ) 10/71 HST ( 0.141988,0.142566 ) 35/246 HST ( 0.142535,0.143181 ) 1/7 EXTENDED HST ( 0.143149,0.143737 ) 35/244 HST ( 0.143457,0.146428 ) 10/69 HST ( 0.328638,0.331754 ) 35/106 HST ( 0.331586,0.335100 ) 1/3 EXTENDED HST ( 0.334928,0.338164 ) 35/104 HST ( 0.984435,1.016066 ) 1/1 EXTENDED HST The supplemental half-space computation shows that these NET maps are rational. SLOPE FUNCTION INFORMATION NUMBER OF FIXED POINTS FOUND: 1 EQUATOR? FIXED POINT c d 0 lambda1 lambda2 lambda1+lambda2 1/0 1 2 No No No No NUMBER OF EQUATORS FOUND: 0 0 0 0 Number of excluded intervals computed by the fixed point finder: 38900 The union of the excluded intervals computed by the fixed point finder became a union of 10,000 disjoint intervals: the search for all slope function fixed points aborted. There are no equators because both elementary divisors are greater than 1. 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. 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=(1,37)(3,35)(4,40)(5,33)(6,38)(7,31)(8,36)(9,29)(10,34)(11,27)(12,32)(13,25)(14,30)(15,23)(16,28)(17,21)(18,26)(20,24)", "b=(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)", "c=(1,39)(2,40)(3,37)(4,38)(5,35)(6,36)(7,33)(8,34)(9,31)(10,32)(11,29)(12,30)(13,27)(14,28)(15,25)(16,26)(17,23)(18,24)(19,21)(20,22)", "d=(1,4)(2,39)(3,6)(5,8)(7,10)(9,12)(11,14)(13,16)(15,18)(17,20)(19,22)(21,24)(23,26)(25,28)(27,30)(29,32)(31,34)(33,36)(35,38)(37,40)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=(1,40)(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)", "b=<1,1,d*b,c^-1,c^-1*b,c^-1,c^-1,c^-1,1,c^-1,1,c^-1,1,c^-1,1,c^-1,1,1,1,1,1,1,1,1,1,1,1,c,1,c,1,c,1,c,c,c,b^-1*c,c,b^-1,c>(3,39)(4,40)(5,37)(6,38)(7,35)(8,36)(9,33)(10,34)(11,31)(12,32)(13,29)(14,30)(15,27)(16,28)(17,25)(18,26)(19,23)(20,24)", "c=<1,1,1,1,1,1,1,1,c,c^-1,c,c^-1,c,c^-1,c,c^-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)", "d=(1,3)(2,40)(4,38)(5,39)(6,36)(7,37)(8,34)(9,35)(10,32)(11,33)(12,30)(13,31)(14,28)(15,29)(16,26)(17,27)(18,24)(19,25)(20,22)(21,23)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,40)(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)", "b=(1,39)(2,40)(3,37)(4,38)(5,35)(6,36)(7,33)(8,34)(9,31)(10,32)(11,29)(12,30)(13,27)(14,28)(15,25)(16,26)(17,23)(18,24)(19,21)(20,22)", "c=(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)", "d=<1,d*b,c^-1,c^-1*b,c^-1,c^-1,c^-1,c^-1,1,c^-1,1,c^-1,1,c^-1,1,c^-1,1,1,1,1,1,1,1,c,1,c,1,c,1,c,1,c,1,c,c,b^-1*c,c,b^-1,c,1>(2,38)(3,39)(4,36)(5,37)(6,34)(7,35)(8,32)(9,33)(10,30)(11,31)(12,28)(13,29)(14,26)(15,27)(16,24)(17,25)(18,22)(19,23)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(1,39)(2,4)(3,37)(5,35)(6,40)(7,33)(8,38)(9,31)(10,36)(11,29)(12,34)(13,27)(14,32)(15,25)(16,30)(17,23)(18,28)(19,21)(20,26)(22,24)", "b=<1,1,1,1,1,1,1,1,c,c^-1,c,c^-1,c,c^-1,c,c^-1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1>(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)", "c=<1,1,d*b,c^-1,c^-1*b,c^-1,c^-1,c^-1,1,c^-1,1,c^-1,1,c^-1,1,c^-1,1,1,1,1,1,1,1,1,1,1,1,c,1,c,1,c,1,c,c,c,b^-1*c,c,b^-1,c>(3,39)(4,40)(5,37)(6,38)(7,35)(8,36)(9,33)(10,34)(11,31)(12,32)(13,29)(14,30)(15,27)(16,28)(17,25)(18,26)(19,23)(20,24)", "d=(1,4)(2,39)(3,6)(5,8)(7,10)(9,12)(11,14)(13,16)(15,18)(17,20)(19,22)(21,24)(23,26)(25,28)(27,30)(29,32)(31,34)(33,36)(35,38)(37,40)", "a*b*c*d");