These Thurston maps are NET maps for every choice of translation term. They have degree 18. They are imprimitive, each factoring as a NET map with degree 3 followed by a Euclidean NET map with degree 6. PURE MODULAR GROUP HURWITZ EQUIVALENCE CLASSES FOR TRANSLATIONS {0} {lambda1} {lambda2} {lambda1+lambda2} Since no Thurston multiplier is 1, this modular group Hurwitz class contains only finitely many Thurston equivalence classes. The number of pure modular group Hurwitz classes in this modular group Hurwitz class is 7. ALL THURSTON MULTIPLIERS c/d IN UNREDUCED FORM 0/3, 1/6, 3/6, 4/3 EXCLUDED INTERVALS FOR THE HALF-SPACE COMPUTATION (-infinity,-0.018302) ( 0.018302,infinity ) 1/0 is the slope of a Thurston obstruction with c = 4 and d = 3. These NET maps are not rational. SLOPE FUNCTION INFORMATION NUMBER OF FIXED POINTS: 4 EQUATOR? FIXED POINT c d 0 lambda1 lambda2 lambda1+lambda2 2/1 1 6 No No No No 1/0 4 3 No No No No -2/1 1 6 No No No No 0/1 3 6 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: 4075 There are no equators because both elementary divisors are greater than 1. No nontrivial cycles were found. Here is the action of the slope function on an invariant set S of slopes. 3N/1 -> 4N/1, (3N+1)/1 -> (4N+2)/1, (3N+2)/1 -> (4N+2)/1 The set S contains infinitely many infinite slope function trajectory tails. Here is the action of the slope function on a set S of slopes. (9N+1)/3 -> (12N+2)/3, (9N+4)/3 -> *, (9N+7)/3 -> *, (9N+2)/3 -> *, (9N+5)/3 -> *, (9N+8)/3 -> (12N+10)/3 Every slope in S either maps to a slope in S or (*) it leaves S, its orbit never to return. The set S contains arbitrarily long segments of slope function trajectories, but it probably contains no infinite slope function trajectories. The slope function maps some slope to the nonslope. If the slope function maps slope s to a slope s' and if the intersection pairing of s with 1/0 is n, then the intersection pairing of s' with 1/0 is at most n. The slope function orbit of every slope whose intersection pairing pairing with 1/0 is at most 50 either ends in the nonslope or ends in one of the slopes described above or it has an infinite tail in one of the infinite sets described above. FUNDAMENTAL GROUP WREATH RECURSIONS When the translation term of the affine map is 0: NewSphereMachine( "a=(2,6)(3,17)(4,16)(5,9)(7,13)(8,12)(11,15)(14,18)", "b=(1,16)(2,3)(4,13)(5,6)(7,10)(8,9)(11,12)(14,15)(17,18)", "c=<1,1,c^-1,1,1,c^-1,1,1,c^-1,1,1,c,1,1,c,1,1,c>(1,2)(3,18)(4,5)(6,15)(7,8)(9,12)(10,11)(13,14)(16,17)", "d=(1,17)(2,4)(5,7)(6,18)(8,10)(9,15)(11,13)(14,16)", "a*b*c*d"); When the translation term of the affine map is lambda1: NewSphereMachine( "a=(1,4)(2,6)(3,17)(5,9)(7,16)(8,12)(10,13)(11,15)(14,18)", "b=(2,3)(4,16)(5,6)(7,13)(8,9)(11,12)(14,15)(17,18)", "c=<1,1,d,1,1,c^-1,1,1,c^-1,1,1,c,1,1,c,1,1,c>(1,2)(4,5)(6,18)(7,8)(9,15)(10,11)(13,14)(16,17)", "d=(1,17)(2,4)(3,6)(5,7)(8,10)(9,18)(11,13)(12,15)(14,16)", "a*b*c*d"); When the translation term of the affine map is lambda2: NewSphereMachine( "a=<1,a*b,c^-1,1,1,c^-1,1,1,c,1,1,c,1,1,c,c*d,1,c*d*c^-1>(1,5)(2,16)(3,15)(4,8)(6,12)(7,11)(10,14)(13,17)", "b=<1,1,c^-1,1,1,c^-1,1,1,c^-1,1,1,c,1,1,c,1,1,c>(1,2)(3,18)(4,5)(6,15)(7,8)(9,12)(10,11)(13,14)(16,17)", "c=(1,16)(2,3)(4,13)(5,6)(7,10)(8,9)(11,12)(14,15)(17,18)", "d=(1,13)(2,18)(3,5)(4,10)(6,8)(9,11)(12,14)(15,17)", "a*b*c*d"); When the translation term of the affine map is lambda1+lambda2: NewSphereMachine( "a=<1,a*b,c^-1,1,1,c^-1,1,1,c^-1,1,1,c,1,1,c,c*d,1,c>(1,5)(2,16)(3,18)(4,8)(6,15)(7,11)(9,12)(10,14)(13,17)", "b=<1,1,d,1,1,c^-1,1,1,c^-1,1,1,c,1,1,c,1,1,c>(1,2)(4,5)(6,18)(7,8)(9,15)(10,11)(13,14)(16,17)", "c=(2,3)(4,16)(5,6)(7,13)(8,9)(11,12)(14,15)(17,18)", "d=(1,16)(2,18)(3,5)(4,13)(6,8)(7,10)(9,11)(12,14)(15,17)", "a*b*c*d");