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): 192 Minimal number of generators: 33 Number of equivalence classes of cusps: 24 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -4/1 -10/3 -3/1 -8/3 -2/1 -3/2 -6/5 0/1 1/1 6/5 3/2 2/1 12/5 5/2 8/3 3/1 36/11 10/3 24/7 4/1 5/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 2/1 -5/1 1/1 2/1 1/0 -4/1 2/1 -7/2 2/1 5/2 3/1 -10/3 2/1 -3/1 1/1 3/1 -14/5 2/1 -11/4 3/1 4/1 1/0 -8/3 2/1 4/1 -13/5 3/1 4/1 1/0 -5/2 3/1 4/1 1/0 -2/1 2/1 1/0 -7/4 3/1 4/1 1/0 -12/7 1/0 -5/3 1/1 2/1 1/0 -18/11 2/1 -13/8 2/1 5/2 3/1 -8/5 2/1 4/1 -11/7 4/1 9/2 5/1 -3/2 1/0 -13/9 -4/1 -7/2 -3/1 -36/25 -3/1 -23/16 -3/1 -8/3 -5/2 -10/7 -2/1 -17/12 -2/1 -3/2 -1/1 -24/17 -1/1 -7/5 -1/1 0/1 1/0 -4/3 0/1 -5/4 1/1 2/1 1/0 -6/5 0/1 2/1 -7/6 1/1 2/1 1/0 -1/1 1/1 2/1 1/0 0/1 1/0 1/1 -1/1 0/1 1/0 6/5 0/1 5/4 0/1 1/2 1/1 4/3 0/1 7/5 0/1 1/2 1/1 10/7 0/1 1/1 3/2 1/2 1/0 14/9 0/1 1/1 11/7 1/1 2/1 1/0 8/5 0/1 2/1 13/8 1/1 2/1 1/0 5/3 1/1 2/1 1/0 2/1 0/1 7/3 1/3 2/5 1/2 12/5 1/2 5/2 1/2 3/5 2/3 18/7 2/3 13/5 2/3 7/10 5/7 8/3 2/3 4/5 11/4 4/5 5/6 1/1 3/1 1/1 13/4 1/1 4/3 3/2 36/11 1/1 23/7 1/1 6/5 5/4 10/3 1/1 4/3 17/5 1/1 4/3 3/2 24/7 1/1 7/2 1/1 4/3 3/2 4/1 2/1 5/1 1/1 2/1 1/0 6/1 0/1 2/1 7/1 1/1 2/1 1/0 1/0 1/1 2/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(13,96,-8,-59) (-6/1,1/0) -> (-18/11,-13/8) Hyperbolic Matrix(23,120,-14,-73) (-6/1,-5/1) -> (-5/3,-18/11) Hyperbolic Matrix(11,48,8,35) (-5/1,-4/1) -> (4/3,7/5) Hyperbolic Matrix(13,48,10,37) (-4/1,-7/2) -> (5/4,4/3) Hyperbolic Matrix(71,240,-50,-169) (-7/2,-10/3) -> (-10/7,-17/12) Hyperbolic Matrix(23,72,-8,-25) (-10/3,-3/1) -> (-3/1,-14/5) Parabolic Matrix(155,432,-108,-301) (-14/5,-11/4) -> (-23/16,-10/7) Hyperbolic Matrix(71,192,44,119) (-11/4,-8/3) -> (8/5,13/8) Hyperbolic Matrix(73,192,46,121) (-8/3,-13/5) -> (11/7,8/5) Hyperbolic Matrix(37,96,-32,-83) (-13/5,-5/2) -> (-7/6,-1/1) Hyperbolic Matrix(11,24,-6,-13) (-5/2,-2/1) -> (-2/1,-7/4) Parabolic 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(119,192,44,71) (-13/8,-8/5) -> (8/3,11/4) Hyperbolic Matrix(121,192,46,73) (-8/5,-11/7) -> (13/5,8/3) Hyperbolic Matrix(47,72,-32,-49) (-11/7,-3/2) -> (-3/2,-13/9) Parabolic Matrix(899,1296,274,395) (-13/9,-36/25) -> (36/11,23/7) Hyperbolic Matrix(901,1296,276,397) (-36/25,-23/16) -> (13/4,36/11) Hyperbolic Matrix(407,576,118,167) (-17/12,-24/17) -> (24/7,7/2) Hyperbolic Matrix(409,576,120,169) (-24/17,-7/5) -> (17/5,24/7) Hyperbolic Matrix(35,48,8,11) (-7/5,-4/3) -> (4/1,5/1) Hyperbolic Matrix(37,48,10,13) (-4/3,-5/4) -> (7/2,4/1) Hyperbolic Matrix(59,72,-50,-61) (-5/4,-6/5) -> (-6/5,-7/6) Parabolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(83,-96,32,-37) (1/1,6/5) -> (18/7,13/5) Hyperbolic Matrix(97,-120,38,-47) (6/5,5/4) -> (5/2,18/7) Hyperbolic Matrix(169,-240,50,-71) (7/5,10/7) -> (10/3,17/5) Hyperbolic Matrix(49,-72,32,-47) (10/7,3/2) -> (3/2,14/9) Parabolic Matrix(277,-432,84,-131) (14/9,11/7) -> (23/7,10/3) Hyperbolic Matrix(59,-96,8,-13) (13/8,5/3) -> (7/1,1/0) Hyperbolic Matrix(13,-24,6,-11) (5/3,2/1) -> (2/1,7/3) Parabolic Matrix(25,-72,8,-23) (11/4,3/1) -> (3/1,13/4) Parabolic Matrix(13,-72,2,-11) (5/1,6/1) -> (6/1,7/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(13,96,-8,-59) -> Matrix(5,-8,2,-3) Matrix(23,120,-14,-73) -> Matrix(1,0,0,1) Matrix(11,48,8,35) -> Matrix(1,-2,2,-3) Matrix(13,48,10,37) -> Matrix(1,-2,0,1) Matrix(71,240,-50,-169) -> Matrix(1,-4,0,1) Matrix(23,72,-8,-25) -> Matrix(1,0,0,1) Matrix(155,432,-108,-301) -> Matrix(5,-12,-2,5) Matrix(71,192,44,119) -> Matrix(1,-2,0,1) Matrix(73,192,46,121) -> Matrix(1,-2,0,1) Matrix(37,96,-32,-83) -> Matrix(1,-2,0,1) Matrix(11,24,-6,-13) -> Matrix(1,0,0,1) Matrix(83,144,34,59) -> Matrix(1,-6,2,-11) Matrix(85,144,36,61) -> Matrix(1,0,2,1) Matrix(119,192,44,71) -> Matrix(3,-10,4,-13) Matrix(121,192,46,73) -> Matrix(3,-10,4,-13) Matrix(47,72,-32,-49) -> Matrix(1,-8,0,1) Matrix(899,1296,274,395) -> Matrix(7,22,6,19) Matrix(901,1296,276,397) -> Matrix(1,4,0,1) Matrix(407,576,118,167) -> Matrix(5,6,4,5) Matrix(409,576,120,169) -> Matrix(3,4,2,3) Matrix(35,48,8,11) -> Matrix(1,2,0,1) Matrix(37,48,10,13) -> Matrix(3,-2,2,-1) Matrix(59,72,-50,-61) -> Matrix(1,0,0,1) Matrix(1,0,2,1) -> Matrix(1,-2,0,1) Matrix(83,-96,32,-37) -> Matrix(7,2,10,3) Matrix(97,-120,38,-47) -> Matrix(5,-2,8,-3) Matrix(169,-240,50,-71) -> Matrix(5,-4,4,-3) Matrix(49,-72,32,-47) -> Matrix(1,0,0,1) Matrix(277,-432,84,-131) -> Matrix(5,-4,4,-3) Matrix(59,-96,8,-13) -> Matrix(1,0,0,1) Matrix(13,-24,6,-11) -> Matrix(1,0,2,1) Matrix(25,-72,8,-23) -> Matrix(9,-8,8,-7) Matrix(13,-72,2,-11) -> Matrix(1,0,0,1) 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): 8 Degree of the the map X: 8 Degree of the the map Y: 32 Permutation triple for Y: ((2,6,7)(3,12,4)(5,10,9)(8,15,14)(11,19,18)(13,22,21)(17,23,31)(26,29,27); (1,4,15,28,18,31,32,29,21,16,5,2)(3,10,27,11)(6,19,24,14,26,30,17,9,25,13,12,20)(7,22,23,8); (1,2,8,24,19,27,32,31,22,25,9,3)(4,13,29,14)(5,17,18,6)(7,20,12,11,28,15,23,30,26,10,16,21)) ----------------------------------------------------------------------- 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 -6/1 2/1 2 2 -4/1 2/1 2 6 -3/1 0 4 -8/3 (2/1,4/1) 0 6 -5/2 0 12 -2/1 0 6 -7/4 0 12 -12/7 1/0 6 2 -5/3 0 12 -18/11 2/1 2 2 -8/5 (2/1,4/1) 0 6 -3/2 1/0 4 4 -4/3 0/1 2 6 -6/5 (0/1,2/1) 0 2 -1/1 0 12 0/1 1/0 2 2 1/1 0 12 6/5 0/1 6 2 4/3 0/1 2 6 3/2 0 4 8/5 (0/1,2/1) 0 6 5/3 0 12 2/1 0/1 2 6 7/3 0 12 12/5 1/2 6 2 5/2 0 12 18/7 2/3 6 2 8/3 (2/3,4/5) 0 6 3/1 1/1 4 4 4/1 2/1 2 6 6/1 (0/1,2/1) 0 2 1/0 0 12 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,48,-3,-29) (-6/1,1/0) -> (-5/3,-18/11) Glide Reflection Matrix(5,24,-1,-5) (-6/1,-4/1) -> (-6/1,-4/1) Reflection Matrix(7,24,5,17) (-4/1,-3/1) -> (4/3,3/2) Glide Reflection Matrix(17,48,11,31) (-3/1,-8/3) -> (3/2,8/5) Glide Reflection Matrix(55,144,-21,-55) (-8/3,-18/7) -> (-8/3,-18/7) Reflection Matrix(19,48,-17,-43) (-13/5,-5/2) -> (-7/6,-1/1) Glide Reflection Matrix(11,24,-6,-13) (-5/2,-2/1) -> (-2/1,-7/4) Parabolic 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(89,144,-55,-89) (-18/11,-8/5) -> (-18/11,-8/5) Reflection Matrix(31,48,11,17) (-8/5,-3/2) -> (8/3,3/1) Glide Reflection Matrix(17,24,5,7) (-3/2,-4/3) -> (3/1,4/1) Glide Reflection Matrix(19,24,-15,-19) (-4/3,-6/5) -> (-4/3,-6/5) Reflection Matrix(41,48,-35,-41) (-6/5,-8/7) -> (-6/5,-8/7) Reflection Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(43,-48,17,-19) (1/1,6/5) -> (5/2,18/7) Glide Reflection Matrix(19,-24,15,-19) (6/5,4/3) -> (6/5,4/3) Reflection Matrix(89,-144,55,-89) (8/5,18/11) -> (8/5,18/11) Reflection Matrix(29,-48,3,-5) (13/8,5/3) -> (7/1,1/0) Glide Reflection Matrix(13,-24,6,-11) (5/3,2/1) -> (2/1,7/3) Parabolic Matrix(55,-144,21,-55) (18/7,8/3) -> (18/7,8/3) Reflection Matrix(5,-24,1,-5) (4/1,6/1) -> (4/1,6/1) Reflection Matrix(7,-48,1,-7) (6/1,8/1) -> (6/1,8/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(5,48,-3,-29) -> Matrix(1,0,1,-1) *** -> (0/1,2/1) Matrix(5,24,-1,-5) -> Matrix(1,0,1,-1) (-6/1,-4/1) -> (0/1,2/1) Matrix(7,24,5,17) -> Matrix(1,-2,1,-3) Matrix(17,48,11,31) -> Matrix(1,-2,1,-3) Matrix(55,144,-21,-55) -> Matrix(3,-8,1,-3) (-8/3,-18/7) -> (2/1,4/1) Matrix(19,48,-17,-43) -> Matrix(1,-2,1,-3) Matrix(11,24,-6,-13) -> Matrix(1,0,0,1) Matrix(83,144,34,59) -> Matrix(1,-6,2,-11) Matrix(85,144,36,61) -> Matrix(1,0,2,1) 0/1 Matrix(89,144,-55,-89) -> Matrix(3,-8,1,-3) (-18/11,-8/5) -> (2/1,4/1) Matrix(31,48,11,17) -> Matrix(1,-6,1,-7) Matrix(17,24,5,7) -> Matrix(1,2,1,1) Matrix(19,24,-15,-19) -> Matrix(1,0,1,-1) (-4/3,-6/5) -> (0/1,2/1) Matrix(41,48,-35,-41) -> Matrix(1,0,1,-1) (-6/5,-8/7) -> (0/1,2/1) Matrix(1,0,2,1) -> Matrix(1,-2,0,1) 1/0 Matrix(43,-48,17,-19) -> Matrix(3,2,5,3) Matrix(19,-24,15,-19) -> Matrix(1,0,1,-1) (6/5,4/3) -> (0/1,2/1) Matrix(89,-144,55,-89) -> Matrix(1,0,1,-1) (8/5,18/11) -> (0/1,2/1) Matrix(29,-48,3,-5) -> Matrix(1,0,1,-1) *** -> (0/1,2/1) Matrix(13,-24,6,-11) -> Matrix(1,0,2,1) 0/1 Matrix(55,-144,21,-55) -> Matrix(11,-8,15,-11) (18/7,8/3) -> (2/3,4/5) Matrix(5,-24,1,-5) -> Matrix(1,0,1,-1) (4/1,6/1) -> (0/1,2/1) Matrix(7,-48,1,-7) -> Matrix(1,0,1,-1) (6/1,8/1) -> (0/1,2/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.