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: 20 Genus: 3 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -9/2 -3/1 -21/8 -12/5 -2/1 -3/2 -6/5 0/1 1/1 3/2 9/5 2/1 5/2 3/1 18/5 15/4 9/2 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 1/0 -11/2 -2/1 1/0 -5/1 1/0 -9/2 -1/1 -4/1 -1/2 -3/1 -1/2 -8/3 -1/2 -21/8 -2/5 -1/3 -34/13 -1/2 -13/5 -3/8 -18/7 -1/3 -5/2 -1/3 0/1 -12/5 -1/2 -7/3 -3/8 -9/4 -1/3 -2/1 -1/4 -3/2 -1/4 0/1 -4/3 -1/4 -17/13 -1/6 -13/10 -1/6 0/1 -9/7 0/1 -5/4 -1/3 0/1 -6/5 -1/4 -1/1 -1/6 0/1 0/1 1/1 1/6 3/2 0/1 1/4 5/3 1/6 12/7 1/4 7/4 0/1 1/3 9/5 0/1 11/6 0/1 1/6 2/1 1/4 9/4 1/3 7/3 3/8 12/5 1/2 5/2 0/1 1/3 3/1 1/2 7/2 2/3 1/1 18/5 1/1 11/3 1/0 15/4 0/1 1/1 19/5 1/0 4/1 1/2 9/2 1/1 5/1 1/0 6/1 1/0 7/1 1/0 8/1 -1/2 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(11,72,-2,-13) (-6/1,1/0) -> (-6/1,-11/2) Parabolic Matrix(47,252,-36,-193) (-11/2,-5/1) -> (-17/13,-13/10) Hyperbolic Matrix(23,108,10,47) (-5/1,-9/2) -> (9/4,7/3) Hyperbolic Matrix(13,54,6,25) (-9/2,-4/1) -> (2/1,9/4) Hyperbolic Matrix(11,36,-4,-13) (-4/1,-3/1) -> (-3/1,-8/3) Parabolic Matrix(335,882,-128,-337) (-8/3,-21/8) -> (-21/8,-34/13) Parabolic Matrix(131,342,18,47) (-34/13,-13/5) -> (7/1,8/1) Hyperbolic Matrix(167,432,46,119) (-13/5,-18/7) -> (18/5,11/3) Hyperbolic Matrix(85,216,24,61) (-18/7,-5/2) -> (7/2,18/5) 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(25,54,6,13) (-9/4,-2/1) -> (4/1,9/2) Hyperbolic Matrix(11,18,-8,-13) (-2/1,-3/2) -> (-3/2,-4/3) Parabolic Matrix(109,144,28,37) (-4/3,-17/13) -> (19/5,4/1) Hyperbolic Matrix(167,216,92,119) (-13/10,-9/7) -> (9/5,11/6) Hyperbolic Matrix(85,108,48,61) (-9/7,-5/4) -> (7/4,9/5) Hyperbolic Matrix(73,90,30,37) (-5/4,-6/5) -> (12/5,5/2) Hyperbolic Matrix(47,54,20,23) (-6/5,-1/1) -> (7/3,12/5) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(13,-18,8,-11) (1/1,3/2) -> (3/2,5/3) Parabolic Matrix(49,-90,6,-11) (11/6,2/1) -> (8/1,1/0) Hyperbolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic Matrix(121,-450,32,-119) (11/3,15/4) -> (15/4,19/5) Parabolic Matrix(13,-72,2,-11) (5/1,6/1) -> (6/1,7/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(11,72,-2,-13) -> Matrix(1,-2,0,1) Matrix(47,252,-36,-193) -> Matrix(1,2,-6,-11) Matrix(23,108,10,47) -> Matrix(3,4,8,11) Matrix(13,54,6,25) -> Matrix(3,2,10,7) Matrix(11,36,-4,-13) -> Matrix(3,2,-8,-5) Matrix(335,882,-128,-337) -> Matrix(1,0,0,1) Matrix(131,342,18,47) -> Matrix(5,2,-8,-3) Matrix(167,432,46,119) -> Matrix(11,4,8,3) Matrix(85,216,24,61) -> Matrix(7,2,10,3) Matrix(59,144,34,83) -> Matrix(1,0,6,1) Matrix(61,144,36,85) -> Matrix(5,2,22,9) Matrix(47,108,10,23) -> Matrix(11,4,8,3) Matrix(25,54,6,13) -> Matrix(7,2,10,3) Matrix(11,18,-8,-13) -> Matrix(1,0,0,1) Matrix(109,144,28,37) -> Matrix(1,0,6,1) Matrix(167,216,92,119) -> Matrix(1,0,12,1) Matrix(85,108,48,61) -> Matrix(1,0,6,1) Matrix(73,90,30,37) -> Matrix(1,0,6,1) Matrix(47,54,20,23) -> Matrix(9,2,22,5) Matrix(1,0,2,1) -> Matrix(1,0,12,1) Matrix(13,-18,8,-11) -> Matrix(1,0,0,1) Matrix(49,-90,6,-11) -> Matrix(1,0,-6,1) Matrix(13,-36,4,-11) -> Matrix(5,-2,8,-3) Matrix(121,-450,32,-119) -> Matrix(1,0,0,1) Matrix(13,-72,2,-11) -> Matrix(1,-2,0,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 18 Minimal number of generators: 4 Number of equivalence classes of cusps: 5 Genus: 0 Degree of H/liftables -> H/(image of liftables): 3 Degree of the the map X: 9 Degree of the the map Y: 24 Permutation triple for Y: ((2,5,13,23,20,9,8,14,6)(3,10,17,7,16,24,22,11,4); (1,4,11,15,14,8,19,24,16,21,20,23,18,17,10,12,5,2)(3,9)(6,7)(13,22); (1,2,7,18,23,22,19,8,3)(6,15,11,13,12,10,9,21,16)) ----------------------------------------------------------------------- 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 6 1 1/1 1/6 1 18 3/2 0 6 5/3 1/6 1 18 12/7 1/4 3 3 7/4 (0/1,1/3) 0 18 9/5 0/1 3 2 11/6 (0/1,1/6) 0 18 2/1 1/4 1 9 9/4 1/3 6 2 7/3 3/8 1 18 12/5 1/2 3 3 5/2 (0/1,1/3) 0 18 3/1 1/2 3 6 7/2 (2/3,1/1) 0 18 18/5 1/1 3 1 11/3 1/0 1 18 15/4 0 6 19/5 1/0 1 18 4/1 1/2 1 9 9/2 1/1 6 2 5/1 1/0 1 18 6/1 1/0 3 3 7/1 1/0 1 18 8/1 -1/2 1 9 1/0 (0/1,1/0) 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(13,-18,8,-11) (1/1,3/2) -> (3/2,5/3) Parabolic 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(71,-126,40,-71) (7/4,9/5) -> (7/4,9/5) Reflection Matrix(109,-198,60,-109) (9/5,11/6) -> (9/5,11/6) Reflection Matrix(49,-90,6,-11) (11/6,2/1) -> (8/1,1/0) Hyperbolic Matrix(25,-54,6,-13) (2/1,9/4) -> (4/1,9/2) Glide 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(121,-450,32,-119) (11/3,15/4) -> (15/4,19/5) Parabolic Matrix(47,-180,6,-23) (19/5,4/1) -> (7/1,8/1) Glide Reflection Matrix(13,-72,2,-11) (5/1,6/1) -> (6/1,7/1) Parabolic IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,2,-1) -> Matrix(1,0,12,-1) (0/1,1/1) -> (0/1,1/6) Matrix(13,-18,8,-11) -> Matrix(1,0,0,1) Matrix(85,-144,36,-61) -> Matrix(9,-2,22,-5) Matrix(83,-144,34,-59) -> Matrix(1,0,6,-1) *** -> (0/1,1/3) Matrix(71,-126,40,-71) -> Matrix(1,0,6,-1) (7/4,9/5) -> (0/1,1/3) Matrix(109,-198,60,-109) -> Matrix(1,0,12,-1) (9/5,11/6) -> (0/1,1/6) Matrix(49,-90,6,-11) -> Matrix(1,0,-6,1) 0/1 Matrix(25,-54,6,-13) -> Matrix(7,-2,10,-3) Matrix(47,-108,10,-23) -> Matrix(11,-4,8,-3) Matrix(13,-36,4,-11) -> Matrix(5,-2,8,-3) 1/2 Matrix(71,-252,20,-71) -> Matrix(5,-4,6,-5) (7/2,18/5) -> (2/3,1/1) Matrix(109,-396,30,-109) -> Matrix(-1,2,0,1) (18/5,11/3) -> (1/1,1/0) Matrix(121,-450,32,-119) -> Matrix(1,0,0,1) Matrix(47,-180,6,-23) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(13,-72,2,-11) -> Matrix(1,-2,0,1) 1/0 ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.