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): 288 Minimal number of generators: 49 Number of equivalence classes of cusps: 32 Genus: 9 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -9/2 -4/1 -3/1 -21/8 -12/5 -24/11 -2/1 -9/5 -3/2 -15/11 -6/5 0/1 1/1 6/5 3/2 21/13 18/11 9/5 2/1 24/11 12/5 5/2 3/1 18/5 15/4 72/19 4/1 9/2 5/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 0/1 -11/2 0/1 1/1 -5/1 0/1 1/1 -9/2 1/0 -13/3 -3/1 1/0 -4/1 -2/1 0/1 -3/1 -1/1 1/0 -8/3 -2/1 0/1 -29/11 -1/1 0/1 -21/8 0/1 1/0 -55/21 -1/1 0/1 -34/13 0/1 -13/5 1/1 1/0 -18/7 1/0 -5/2 -2/1 1/0 -12/5 -2/1 0/1 -7/3 -1/1 0/1 -9/4 1/0 -11/5 -3/1 1/0 -24/11 -4/1 -2/1 -13/6 -4/1 -3/1 -2/1 -2/1 -13/7 -3/2 -1/1 -24/13 -2/1 -4/3 -11/6 -2/1 -1/1 -9/5 -3/2 1/0 -7/4 -2/1 -3/2 -12/7 -2/1 -4/3 -5/3 -4/3 -1/1 -3/2 -2/1 -1/1 -7/5 -4/3 -1/1 -18/13 -1/1 -11/8 -1/1 -2/3 -26/19 0/1 -15/11 -1/1 0/1 -34/25 0/1 -53/39 -1/3 0/1 -72/53 0/1 -19/14 0/1 1/0 -4/3 -2/1 0/1 -17/13 -2/1 -1/1 -13/10 -2/1 -1/1 -9/7 -3/2 1/0 -5/4 -2/1 1/0 -6/5 -2/1 -7/6 -2/1 -3/2 -8/7 -2/1 -8/5 -1/1 -3/2 -1/1 0/1 -1/1 1/1 -1/1 -3/4 6/5 -2/3 11/9 -3/5 -1/2 5/4 -2/3 -1/2 9/7 -3/4 -1/2 13/10 -1/1 -2/3 4/3 -2/3 0/1 3/2 -1/1 -2/3 8/5 -2/3 0/1 29/18 -1/2 0/1 21/13 -1/1 0/1 55/34 -1/2 0/1 34/21 0/1 13/8 -2/1 -1/1 18/11 -1/1 5/3 -1/1 -4/5 12/7 -4/5 -2/3 7/4 -3/4 -2/3 9/5 -3/4 -1/2 11/6 -1/1 -2/3 24/13 -4/5 -2/3 13/7 -1/1 -3/4 2/1 -2/3 13/6 -3/5 -4/7 24/11 -2/3 -4/7 11/5 -3/5 -1/2 9/4 -1/2 7/3 -1/1 0/1 12/5 -2/3 0/1 5/2 -2/3 -1/2 3/1 -1/1 -1/2 7/2 -2/3 -1/2 18/5 -1/2 11/3 -1/2 -1/3 26/7 0/1 15/4 -1/2 0/1 34/9 0/1 53/14 -1/4 0/1 72/19 0/1 19/5 -1/1 0/1 4/1 -2/3 0/1 17/4 -2/3 -1/2 13/3 -3/5 -1/2 9/2 -1/2 5/1 -1/3 0/1 6/1 0/1 7/1 -1/1 0/1 8/1 -2/1 0/1 1/0 -1/1 0/1 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(49,216,22,97) (-9/2,-13/3) -> (11/5,9/4) Hyperbolic Matrix(25,108,-22,-95) (-13/3,-4/1) -> (-8/7,-1/1) Hyperbolic Matrix(11,36,-4,-13) (-4/1,-3/1) -> (-3/1,-8/3) Parabolic Matrix(109,288,14,37) (-8/3,-29/11) -> (7/1,8/1) Hyperbolic Matrix(671,1764,-256,-673) (-29/11,-21/8) -> (-21/8,-55/21) Parabolic Matrix(1403,3672,-1032,-2701) (-55/21,-34/13) -> (-34/25,-53/39) Hyperbolic Matrix(179,468,96,251) (-34/13,-13/5) -> (13/7,2/1) 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(97,216,22,49) (-9/4,-11/5) -> (13/3,9/2) Hyperbolic Matrix(263,576,142,311) (-11/5,-24/11) -> (24/13,13/7) Hyperbolic Matrix(265,576,144,313) (-24/11,-13/6) -> (11/6,24/13) Hyperbolic Matrix(217,468,134,289) (-13/6,-2/1) -> (34/21,13/8) Hyperbolic Matrix(193,360,52,97) (-2/1,-13/7) -> (11/3,26/7) Hyperbolic Matrix(311,576,142,263) (-13/7,-24/13) -> (24/11,11/5) Hyperbolic Matrix(313,576,144,265) (-24/13,-11/6) -> (13/6,24/11) Hyperbolic Matrix(119,216,92,167) (-11/6,-9/5) -> (9/7,13/10) 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(155,216,94,131) (-7/5,-18/13) -> (18/11,5/3) Hyperbolic Matrix(313,432,192,265) (-18/13,-11/8) -> (13/8,18/11) Hyperbolic Matrix(263,360,122,167) (-11/8,-26/19) -> (2/1,13/6) Hyperbolic Matrix(659,900,-484,-661) (-26/19,-15/11) -> (-15/11,-34/25) Parabolic Matrix(3815,5184,1006,1367) (-53/39,-72/53) -> (72/19,19/5) Hyperbolic Matrix(3817,5184,1008,1369) (-72/53,-19/14) -> (53/14,72/19) Hyperbolic Matrix(107,144,26,35) (-19/14,-4/3) -> (4/1,17/4) Hyperbolic 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(59,72,-50,-61) (-5/4,-6/5) -> (-6/5,-7/6) Parabolic Matrix(251,288,156,179) (-7/6,-8/7) -> (8/5,29/18) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(61,-72,50,-59) (1/1,6/5) -> (6/5,11/9) Parabolic Matrix(205,-252,48,-59) (11/9,5/4) -> (17/4,13/3) Hyperbolic Matrix(83,-108,10,-13) (13/10,4/3) -> (8/1,1/0) Hyperbolic Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(1093,-1764,676,-1091) (29/18,21/13) -> (21/13,55/34) Parabolic Matrix(2269,-3672,600,-971) (55/34,34/21) -> (34/9,53/14) Hyperbolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic Matrix(241,-900,64,-239) (26/7,15/4) -> (15/4,34/9) 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(1,0,2,1) Matrix(47,252,-36,-193) -> Matrix(1,-2,0,1) Matrix(23,108,10,47) -> Matrix(1,0,-2,1) Matrix(49,216,22,97) -> Matrix(1,6,-2,-11) Matrix(25,108,-22,-95) -> Matrix(3,8,-2,-5) Matrix(11,36,-4,-13) -> Matrix(1,0,0,1) Matrix(109,288,14,37) -> Matrix(1,0,0,1) Matrix(671,1764,-256,-673) -> Matrix(1,0,0,1) Matrix(1403,3672,-1032,-2701) -> Matrix(1,0,-2,1) Matrix(179,468,96,251) -> Matrix(3,-2,-4,3) Matrix(167,432,46,119) -> Matrix(1,-2,-2,5) Matrix(85,216,24,61) -> Matrix(1,4,-2,-7) Matrix(59,144,34,83) -> Matrix(3,4,-4,-5) Matrix(61,144,36,85) -> Matrix(3,4,-4,-5) Matrix(47,108,10,23) -> Matrix(1,0,-2,1) Matrix(97,216,22,49) -> Matrix(1,6,-2,-11) Matrix(263,576,142,311) -> Matrix(3,10,-4,-13) Matrix(265,576,144,313) -> Matrix(3,10,-4,-13) Matrix(217,468,134,289) -> Matrix(1,2,0,1) Matrix(193,360,52,97) -> Matrix(1,2,-4,-7) Matrix(311,576,142,263) -> Matrix(7,10,-12,-17) Matrix(313,576,144,265) -> Matrix(7,10,-12,-17) Matrix(119,216,92,167) -> Matrix(3,4,-4,-5) Matrix(61,108,48,85) -> Matrix(3,4,-4,-5) Matrix(83,144,34,59) -> Matrix(3,4,-4,-5) Matrix(85,144,36,61) -> Matrix(3,4,-4,-5) Matrix(23,36,-16,-25) -> Matrix(1,0,0,1) Matrix(155,216,94,131) -> Matrix(7,8,-8,-9) Matrix(313,432,192,265) -> Matrix(5,4,-4,-3) Matrix(263,360,122,167) -> Matrix(5,2,-8,-3) Matrix(659,900,-484,-661) -> Matrix(1,0,0,1) Matrix(3815,5184,1006,1367) -> Matrix(1,0,2,1) Matrix(3817,5184,1008,1369) -> Matrix(1,0,-4,1) Matrix(107,144,26,35) -> Matrix(1,2,-2,-3) Matrix(109,144,28,37) -> Matrix(1,2,-2,-3) Matrix(167,216,92,119) -> Matrix(3,4,-4,-5) Matrix(85,108,48,61) -> Matrix(3,4,-4,-5) Matrix(59,72,-50,-61) -> Matrix(3,8,-2,-5) Matrix(251,288,156,179) -> Matrix(1,2,-4,-7) Matrix(1,0,2,1) -> Matrix(5,6,-6,-7) Matrix(61,-72,50,-59) -> Matrix(11,8,-18,-13) Matrix(205,-252,48,-59) -> Matrix(1,0,0,1) Matrix(83,-108,10,-13) -> Matrix(3,2,-2,-1) Matrix(25,-36,16,-23) -> Matrix(1,0,0,1) Matrix(1093,-1764,676,-1091) -> Matrix(1,0,0,1) Matrix(2269,-3672,600,-971) -> Matrix(1,0,-2,1) Matrix(13,-36,4,-11) -> Matrix(1,0,0,1) Matrix(241,-900,64,-239) -> Matrix(1,0,0,1) Matrix(13,-72,2,-11) -> Matrix(1,0,2,1) 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): 9 Degree of the the map X: 9 Degree of the the map Y: 48 Permutation triple for Y: ((2,6,19,39,43,46,40,20,7)(3,11,29,38,47,41,30,12,4)(5,16,36,27,10,9,26,37,17)(8,22,42,33,14,13,32,35,23); (1,4,14,34,26,40,45,41,22,28,27,39,44,29,35,15,5,2)(3,10)(6,17,31,13,12,21,20,9,25,42,47,48,43,36,24,23,11,18)(7,8)(16,30)(19,33)(32,46)(37,38); (1,2,8,24,36,30,45,40,32,31,17,38,44,39,33,25,9,3)(4,13)(5,6)(7,21,12,16,15,35,46,48,47,37,34,14,19,18,11,10,28,22)(20,26)(23,29)(27,43)(41,42)) ----------------------------------------------------------------------- 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 -1/1 3 1 1/1 (-1/1,-3/4) 0 18 6/5 -2/3 3 3 11/9 (-3/5,-1/2) 0 18 5/4 (-2/3,-1/2) 0 18 9/7 0 2 13/10 (-1/1,-2/3) 0 18 4/3 0 9 3/2 0 6 8/5 0 9 29/18 (-1/2,0/1) 0 18 21/13 0 6 55/34 (-1/2,0/1) 0 18 34/21 0/1 1 9 13/8 (-2/1,-1/1) 0 18 18/11 -1/1 6 1 5/3 (-1/1,-4/5) 0 18 12/7 0 3 7/4 (-3/4,-2/3) 0 18 9/5 0 2 11/6 (-1/1,-2/3) 0 18 24/13 0 3 13/7 (-1/1,-3/4) 0 18 2/1 -2/3 1 9 13/6 (-3/5,-4/7) 0 18 24/11 0 3 11/5 (-3/5,-1/2) 0 18 9/4 -1/2 3 2 7/3 (-1/1,0/1) 0 18 12/5 0 3 5/2 (-2/3,-1/2) 0 18 3/1 0 6 7/2 (-2/3,-1/2) 0 18 18/5 -1/2 3 1 11/3 (-1/2,-1/3) 0 18 26/7 0/1 1 9 15/4 0 6 34/9 0/1 1 9 53/14 (-1/4,0/1) 0 18 72/19 0/1 3 1 19/5 (-1/1,0/1) 0 18 4/1 0 9 17/4 (-2/3,-1/2) 0 18 13/3 (-3/5,-1/2) 0 18 9/2 -1/2 3 2 5/1 (-1/3,0/1) 0 18 6/1 0/1 3 3 7/1 (-1/1,0/1) 0 18 8/1 0 9 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(61,-72,50,-59) (1/1,6/5) -> (6/5,11/9) Parabolic Matrix(205,-252,48,-59) (11/9,5/4) -> (17/4,13/3) Hyperbolic Matrix(85,-108,48,-61) (5/4,9/7) -> (7/4,9/5) Glide Reflection Matrix(167,-216,92,-119) (9/7,13/10) -> (9/5,11/6) Glide Reflection Matrix(83,-108,10,-13) (13/10,4/3) -> (8/1,1/0) Hyperbolic Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(157,-252,38,-61) (8/5,29/18) -> (4/1,17/4) Glide Reflection Matrix(1093,-1764,676,-1091) (29/18,21/13) -> (21/13,55/34) Parabolic Matrix(2269,-3672,600,-971) (55/34,34/21) -> (34/9,53/14) Hyperbolic Matrix(289,-468,134,-217) (34/21,13/8) -> (2/1,13/6) Glide Reflection Matrix(287,-468,176,-287) (13/8,18/11) -> (13/8,18/11) Reflection Matrix(109,-180,66,-109) (18/11,5/3) -> (18/11,5/3) Reflection 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(313,-576,144,-265) (11/6,24/13) -> (13/6,24/11) Glide Reflection Matrix(311,-576,142,-263) (24/13,13/7) -> (24/11,11/5) Glide Reflection Matrix(193,-360,52,-97) (13/7,2/1) -> (11/3,26/7) Glide Reflection Matrix(97,-216,22,-49) (11/5,9/4) -> (13/3,9/2) 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(241,-900,64,-239) (26/7,15/4) -> (15/4,34/9) Parabolic Matrix(2015,-7632,532,-2015) (53/14,72/19) -> (53/14,72/19) Reflection Matrix(721,-2736,190,-721) (72/19,19/5) -> (72/19,19/5) Reflection 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(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) Matrix(1,0,2,-1) -> Matrix(7,6,-8,-7) (0/1,1/1) -> (-1/1,-3/4) Matrix(61,-72,50,-59) -> Matrix(11,8,-18,-13) -2/3 Matrix(205,-252,48,-59) -> Matrix(1,0,0,1) Matrix(85,-108,48,-61) -> Matrix(5,4,-6,-5) *** -> (-1/1,-2/3) Matrix(167,-216,92,-119) -> Matrix(5,4,-6,-5) *** -> (-1/1,-2/3) Matrix(83,-108,10,-13) -> Matrix(3,2,-2,-1) -1/1 Matrix(25,-36,16,-23) -> Matrix(1,0,0,1) Matrix(157,-252,38,-61) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(1093,-1764,676,-1091) -> Matrix(1,0,0,1) Matrix(2269,-3672,600,-971) -> Matrix(1,0,-2,1) 0/1 Matrix(289,-468,134,-217) -> Matrix(1,-2,-2,3) Matrix(287,-468,176,-287) -> Matrix(3,4,-2,-3) (13/8,18/11) -> (-2/1,-1/1) Matrix(109,-180,66,-109) -> Matrix(9,8,-10,-9) (18/11,5/3) -> (-1/1,-4/5) Matrix(85,-144,36,-61) -> Matrix(5,4,-6,-5) *** -> (-1/1,-2/3) Matrix(83,-144,34,-59) -> Matrix(5,4,-6,-5) *** -> (-1/1,-2/3) Matrix(313,-576,144,-265) -> Matrix(13,10,-22,-17) Matrix(311,-576,142,-263) -> Matrix(13,10,-22,-17) Matrix(193,-360,52,-97) -> Matrix(3,2,-10,-7) Matrix(97,-216,22,-49) -> Matrix(11,6,-20,-11) *** -> (-3/5,-1/2) Matrix(47,-108,10,-23) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(13,-36,4,-11) -> Matrix(1,0,0,1) Matrix(71,-252,20,-71) -> Matrix(7,4,-12,-7) (7/2,18/5) -> (-2/3,-1/2) Matrix(109,-396,30,-109) -> Matrix(5,2,-12,-5) (18/5,11/3) -> (-1/2,-1/3) Matrix(241,-900,64,-239) -> Matrix(1,0,0,1) Matrix(2015,-7632,532,-2015) -> Matrix(-1,0,8,1) (53/14,72/19) -> (-1/4,0/1) Matrix(721,-2736,190,-721) -> Matrix(-1,0,2,1) (72/19,19/5) -> (-1/1,0/1) Matrix(47,-180,6,-23) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(13,-72,2,-11) -> Matrix(1,0,2,1) 0/1 ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.