These Thurston maps are NET maps for every choice of translation term. They are primitive and have degree 39. 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 0/13, 0/39, 1/39, 1/13, 1/3, 2/3, 1/1, 3/3, 4/3, 5/3, 2/1, 6/3, 7/3, 3/1 10/3, 11/3, 4/1, 5/1, 6/1, 7/1, 8/1, 9/1, 10/1, 11/1, 12/1, 14/1, 15/1, 16/1 17/1, 18/1, 19/1, 24/1, 25/1, 27/1, 29/1, 33/1, 36/1, 37/1 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,-1.000000) (-0.985201,-0.500000) (-0.492393,-0.338969) (-0.327289,-0.016854) ( 0.014495,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 (-1.004140,-0.995894) -1/1 EXTENDED HST (-0.996554,-0.993925) -209/210 HST (-0.995219,-0.993420) -175/176 HST (-0.994288,-0.992212) -148/149 HST (-0.993244,-0.990620) -123/124 HST (-0.991875,-0.990984) -116/117 HST (-0.991402,-0.988201) -97/98 HST (-0.989718,-0.985907) -81/82 HST (-0.987696,-0.987612) -80/81 HST (-0.987649,-0.982939) -67/68 HST (-0.503185,-0.496855) -1/2 EXTENDED HST (-0.498371,-0.495268) -78/157 HST (-0.496777,-0.487758) -33/67 HST (-0.339539,-0.337872) -21/62 HST (-0.338605,-0.337084) -25/74 HST (-0.337611,-0.336324) -30/89 HST (-0.336501,-0.335880) -38/113 HST (-0.336212,-0.335428) -44/131 HST (-0.335701,-0.335034) -54/161 HST (-0.335273,-0.334729) -66/197 HST (-0.334936,-0.334487) -80/239 HST (-0.334725,-0.331987) -1/3 EXTENDED HST (-0.332259,-0.329253) -44/133 HST (-0.330819,-0.329900) -38/115 HST (-0.330365,-0.329205) -32/97 HST (-0.329892,-0.328654) -28/85 HST (-0.329297,-0.327859) -23/70 HST (-0.328533,-0.326629) -19/58 HST (-0.017722,-0.015983) -2/119 HST (-0.016744,-0.016590) -1/60 HST (-0.017947,-0.013256) -1/63 HST (-0.015201,-0.011098) -1/76 HST (-0.011214,-0.010773) -1/91 HST (-0.012300,-0.008965) -1/93 HST (-0.010241,-0.007586) -1/112 HST (-0.007592,-0.007560) -1/132 HST (-0.008658,-0.006268) -1/133 HST (-0.007222,-0.005270) -1/160 HST (-0.006065,-0.004450) -1/190 HST (-0.005112,-0.003732) -1/225 HST (-0.004072,0.004140 ) 0/1 EXTENDED HST ( 0.003950,0.005489 ) 1/211 HST ( 0.004756,0.004768 ) 1/210 HST ( 0.004772,0.006592 ) 1/175 HST ( 0.005727,0.007981 ) 1/147 HST ( 0.006813,0.009610 ) 1/123 HST ( 0.008130,0.009009 ) 1/117 HST ( 0.008570,0.011838 ) 1/98 HST ( 0.010242,0.014148 ) 1/82 HST ( 0.012236,0.017223 ) 1/69 HST The supplemental half-space computation shows that these NET maps are rational. SLOPE FUNCTION INFORMATION NUMBER OF FIXED POINTS: 20 EQUATOR? FIXED POINT c d 0 lambda1 lambda2 lambda1+lambda2 2/1 1 39 No No Yes Yes 4/1 1 39 No No Yes Yes 38/1 1 39 No No Yes Yes 38/3 1 39 No No Yes Yes 38/5 1 39 No No Yes Yes 38/7 1 39 No No Yes Yes 38/9 1 39 No No Yes Yes 38/11 1 39 No No Yes Yes 38/13 1 39 No No Yes Yes 38/15 1 39 No No Yes Yes 38/17 1 39 No No Yes Yes 38/21 1 39 No No Yes Yes 38/23 1 39 No No Yes Yes 38/25 1 39 No No Yes Yes 38/27 1 39 No No Yes Yes 38/29 1 39 No No Yes Yes 38/31 1 39 No No Yes Yes 38/33 1 39 No No Yes Yes 38/35 1 39 No No Yes Yes 38/37 1 39 No No Yes Yes NUMBER OF EQUATORS: 0 0 20 20 There are no more slope function fixed points. Number of excluded intervals computed by the fixed point finder: 10661 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 |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=(1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)", "b=(1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)", "c=(1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)", "d=(1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)", "b=<1,b,c^-1*b,c^-1*b,c^-1*b,c^-1*b,c^-1*b,c^-1*b,c^-1*b,c^-1*b,b,b,b,b,b,b,b,b,b,b,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1*c,b^-1*c,b^-1*c,b^-1*c,b^-1*c,b^-1*c,b^-1*c,b^-1*c,b^-1*c>(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)", "c=(1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)", "d=(1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=(1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)", "b=(1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)", "c=(1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)", "d=(1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=(1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)", "b=(1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)", "c=<1,b,c^-1*b,c^-1*b,c^-1*b,c^-1*b,c^-1*b,c^-1*b,c^-1*b,c^-1*b,b,b,b,b,b,b,b,b,b,b,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1,b^-1*c,b^-1*c,b^-1*c,b^-1*c,b^-1*c,b^-1*c,b^-1*c,b^-1*c,b^-1*c>(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)", "d=(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)", "a*b*c*d"); ****************************INTEGER OVERFLOW REPORT***************************** Imminent integer overflow occurred during the computation of a fixed point excluded interval about -64039/131641. Imminent integer overflow halted computation of the slope function at slope -1493241/866258 during the modular group computation. This caused the modular group computation to abort.