INFORMATION ON THE IMAGE IN PSL(2,Z) OF THE GROUP OF PURE MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE Index in PSL(2,Z): 144 Minimal number of generators: 25 Number of equivalence classes of cusps: 16 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -3/1 -12/5 -3/2 0/1 1/1 3/2 9/5 2/1 12/5 3/1 18/5 4/1 9/2 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -1/1 0/1 -5/1 1/0 -9/2 -1/1 -4/1 -1/2 -3/1 -1/2 -8/3 -1/2 -5/2 -1/3 0/1 -12/5 -1/2 -7/3 -1/2 -9/4 -1/3 -2/1 -1/3 0/1 -9/5 0/1 -7/4 -1/1 0/1 -12/7 -1/2 -5/3 -1/2 -3/2 -1/3 0/1 -7/5 -1/2 -18/13 -1/3 -11/8 -1/3 -2/7 -4/3 -1/4 -9/7 0/1 -5/4 -1/3 0/1 -6/5 -1/3 0/1 -1/1 -1/4 0/1 0/1 1/1 1/2 6/5 0/1 1/1 5/4 0/1 1/1 9/7 0/1 4/3 1/2 3/2 0/1 1/1 8/5 1/2 5/3 1/0 12/7 1/0 7/4 -1/1 0/1 9/5 0/1 2/1 0/1 1/1 9/4 1/1 7/3 1/0 12/5 1/0 5/2 0/1 1/1 3/1 1/0 7/2 -2/1 -1/1 18/5 -1/1 11/3 -1/2 4/1 1/0 9/2 -1/1 5/1 -1/2 6/1 -1/1 0/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,36,4,29) (-6/1,1/0) -> (6/5,5/4) Hyperbolic Matrix(7,36,6,31) (-6/1,-5/1) -> (1/1,6/5) Hyperbolic Matrix(23,108,10,47) (-5/1,-9/2) -> (9/4,7/3) Hyperbolic Matrix(17,72,4,17) (-9/2,-4/1) -> (4/1,9/2) Hyperbolic Matrix(11,36,-4,-13) (-4/1,-3/1) -> (-3/1,-8/3) Parabolic Matrix(41,108,-30,-79) (-8/3,-5/2) -> (-11/8,-4/3) Hyperbolic Matrix(59,144,34,83) (-5/2,-12/5) -> (12/7,7/4) Hyperbolic Matrix(61,144,36,85) (-12/5,-7/3) -> (5/3,12/7) Hyperbolic Matrix(47,108,10,23) (-7/3,-9/4) -> (9/2,5/1) Hyperbolic Matrix(17,36,8,17) (-9/4,-2/1) -> (2/1,9/4) Hyperbolic Matrix(19,36,10,19) (-2/1,-9/5) -> (9/5,2/1) Hyperbolic Matrix(61,108,48,85) (-9/5,-7/4) -> (5/4,9/7) Hyperbolic Matrix(83,144,34,59) (-7/4,-12/7) -> (12/5,5/2) Hyperbolic Matrix(85,144,36,61) (-12/7,-5/3) -> (7/3,12/5) Hyperbolic Matrix(23,36,-16,-25) (-5/3,-3/2) -> (-3/2,-7/5) Parabolic Matrix(233,324,64,89) (-7/5,-18/13) -> (18/5,11/3) Hyperbolic Matrix(235,324,66,91) (-18/13,-11/8) -> (7/2,18/5) Hyperbolic Matrix(55,72,42,55) (-4/3,-9/7) -> (9/7,4/3) Hyperbolic Matrix(85,108,48,61) (-9/7,-5/4) -> (7/4,9/5) Hyperbolic Matrix(29,36,4,5) (-5/4,-6/5) -> (6/1,1/0) Hyperbolic Matrix(31,36,6,7) (-6/5,-1/1) -> (5/1,6/1) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(67,-108,18,-29) (8/5,5/3) -> (11/3,4/1) Hyperbolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(5,36,4,29) -> Matrix(1,0,2,1) Matrix(7,36,6,31) -> Matrix(1,0,2,1) Matrix(23,108,10,47) -> Matrix(1,2,0,1) Matrix(17,72,4,17) -> Matrix(3,2,-2,-1) Matrix(11,36,-4,-13) -> Matrix(3,2,-8,-5) Matrix(41,108,-30,-79) -> Matrix(5,2,-18,-7) Matrix(59,144,34,83) -> Matrix(1,0,2,1) Matrix(61,144,36,85) -> Matrix(5,2,2,1) Matrix(47,108,10,23) -> Matrix(5,2,-8,-3) Matrix(17,36,8,17) -> Matrix(1,0,4,1) Matrix(19,36,10,19) -> Matrix(1,0,4,1) Matrix(61,108,48,85) -> Matrix(1,0,2,1) Matrix(83,144,34,59) -> Matrix(1,0,2,1) Matrix(85,144,36,61) -> Matrix(5,2,2,1) Matrix(23,36,-16,-25) -> Matrix(1,0,0,1) Matrix(233,324,64,89) -> Matrix(5,2,-8,-3) Matrix(235,324,66,91) -> Matrix(13,4,-10,-3) Matrix(55,72,42,55) -> Matrix(1,0,6,1) Matrix(85,108,48,61) -> Matrix(1,0,2,1) Matrix(29,36,4,5) -> Matrix(1,0,2,1) Matrix(31,36,6,7) -> Matrix(1,0,2,1) Matrix(1,0,2,1) -> Matrix(1,0,6,1) Matrix(25,-36,16,-23) -> Matrix(1,0,0,1) Matrix(67,-108,18,-29) -> Matrix(1,0,-2,1) Matrix(13,-36,4,-11) -> Matrix(1,-2,0,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 6 Minimal number of generators: 2 Number of equivalence classes of cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 6 Degree of the the map X: 6 Degree of the the map Y: 24 Permutation triple for Y: ((2,6,19,12,4,3,11,20,7)(5,16,14,13,10,9,22,8,17); (1,4,14,23,22,11,18,6,17,24,13,12,21,20,9,15,5,2)(3,10)(7,8)(16,19); (1,2,8,23,14,19,18,11,10,24,17,7,21,12,16,15,9,3)(4,13)(5,6)(20,22)) ----------------------------------------------------------------------- Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift elements of DeckMod(f) via pi_1: 0 DeckMod(f) is trivial. Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift modular group liftables via pi_1: 0 The subgroup of modular group liftables which arise from translations is trivial. ----------------------------------------------------------------------- The image of the modular group liftables in PSL(2,Z) equals the image of the pure modular group liftables. ----------------------------------------------------------------------- INFORMATION ON THE IMAGE IN PGL(2,Z) OF THE GROUP OF EXTENDED MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE c d 0/1 0/1 3 1 1/1 1/2 1 18 6/5 0 3 5/4 (0/1,1/1) 0 18 9/7 0/1 3 2 4/3 1/2 1 9 3/2 0 6 8/5 1/2 1 9 5/3 1/0 1 18 12/7 1/0 3 3 7/4 (-1/1,0/1) 0 18 9/5 0/1 3 2 2/1 (0/1,1/1) 0 9 9/4 1/1 3 2 7/3 1/0 1 18 12/5 1/0 3 3 5/2 (0/1,1/1) 0 18 3/1 1/0 3 6 7/2 (-2/1,-1/1) 0 18 18/5 -1/1 3 1 11/3 -1/2 1 18 4/1 1/0 1 9 9/2 -1/1 3 2 5/1 -1/2 1 18 6/1 0 3 1/0 (-1/1,0/1) 0 18 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Reflection Matrix(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(31,-36,6,-7) (1/1,6/5) -> (5/1,6/1) Glide Reflection Matrix(29,-36,4,-5) (6/5,5/4) -> (6/1,1/0) Glide Reflection Matrix(85,-108,48,-61) (5/4,9/7) -> (7/4,9/5) Glide Reflection Matrix(55,-72,42,-55) (9/7,4/3) -> (9/7,4/3) Reflection Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(67,-108,18,-29) (8/5,5/3) -> (11/3,4/1) Hyperbolic Matrix(85,-144,36,-61) (5/3,12/7) -> (7/3,12/5) Glide Reflection Matrix(83,-144,34,-59) (12/7,7/4) -> (12/5,5/2) Glide Reflection Matrix(19,-36,10,-19) (9/5,2/1) -> (9/5,2/1) Reflection Matrix(17,-36,8,-17) (2/1,9/4) -> (2/1,9/4) Reflection Matrix(47,-108,10,-23) (9/4,7/3) -> (9/2,5/1) Glide Reflection Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic Matrix(71,-252,20,-71) (7/2,18/5) -> (7/2,18/5) Reflection Matrix(109,-396,30,-109) (18/5,11/3) -> (18/5,11/3) Reflection Matrix(17,-72,4,-17) (4/1,9/2) -> (4/1,9/2) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) Matrix(1,0,2,-1) -> Matrix(1,0,4,-1) (0/1,1/1) -> (0/1,1/2) Matrix(31,-36,6,-7) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(29,-36,4,-5) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(85,-108,48,-61) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(55,-72,42,-55) -> Matrix(1,0,4,-1) (9/7,4/3) -> (0/1,1/2) Matrix(25,-36,16,-23) -> Matrix(1,0,0,1) Matrix(67,-108,18,-29) -> Matrix(1,0,-2,1) 0/1 Matrix(85,-144,36,-61) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(83,-144,34,-59) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(19,-36,10,-19) -> Matrix(1,0,2,-1) (9/5,2/1) -> (0/1,1/1) Matrix(17,-36,8,-17) -> Matrix(1,0,2,-1) (2/1,9/4) -> (0/1,1/1) Matrix(47,-108,10,-23) -> Matrix(1,-2,-2,3) Matrix(13,-36,4,-11) -> Matrix(1,-2,0,1) 1/0 Matrix(71,-252,20,-71) -> Matrix(3,4,-2,-3) (7/2,18/5) -> (-2/1,-1/1) Matrix(109,-396,30,-109) -> Matrix(3,2,-4,-3) (18/5,11/3) -> (-1/1,-1/2) Matrix(17,-72,4,-17) -> Matrix(1,2,0,-1) (4/1,9/2) -> (-1/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.