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 0/1 -5/1 -1/1 -4/1 0/1 -7/2 0/1 1/0 -10/3 0/1 1/1 1/0 -3/1 1/0 -14/5 -2/1 -1/1 1/0 -11/4 -1/1 0/1 -8/3 1/0 -13/5 -1/1 -5/2 0/1 1/0 -2/1 -2/1 -1/1 1/0 -7/4 0/1 1/0 -12/7 1/0 -5/3 -3/1 -18/11 -2/1 -13/8 -2/1 -1/1 -8/5 1/0 -11/7 -3/1 -3/2 -2/1 -13/9 -1/1 -36/25 -1/1 -23/16 -1/1 0/1 -10/7 -2/1 -1/1 1/0 -17/12 -2/1 1/0 -24/17 -2/1 -7/5 -1/1 -4/3 -2/1 -5/4 -2/1 -3/2 -6/5 -2/1 -4/3 -7/6 -2/1 -3/2 -1/1 -1/1 0/1 -1/1 1/1 -1/1 6/5 -4/5 -2/3 5/4 -3/4 -2/3 4/3 -2/3 7/5 -1/1 10/7 -1/1 -2/3 -1/2 3/2 -2/3 14/9 -2/3 -5/8 -3/5 11/7 -3/5 8/5 -1/2 13/8 -1/1 -2/3 5/3 -3/5 2/1 -1/1 -2/3 -1/2 7/3 -3/5 12/5 -1/2 5/2 -1/2 0/1 18/7 -2/3 0/1 13/5 -1/1 8/3 -1/2 11/4 -1/1 0/1 3/1 -1/2 13/4 -2/5 -1/3 36/11 -1/3 23/7 -1/3 10/3 -1/2 -1/3 0/1 17/5 -1/3 24/7 0/1 7/2 -1/2 0/1 4/1 0/1 5/1 -1/1 6/1 0/1 7/1 1/1 1/0 -1/1 0/1 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(3,2,-2,-1) Matrix(23,120,-14,-73) -> Matrix(1,-2,0,1) Matrix(11,48,8,35) -> Matrix(1,2,-2,-3) Matrix(13,48,10,37) -> Matrix(3,-2,-4,3) Matrix(71,240,-50,-169) -> Matrix(1,-2,0,1) Matrix(23,72,-8,-25) -> Matrix(1,-2,0,1) Matrix(155,432,-108,-301) -> Matrix(1,0,0,1) Matrix(71,192,44,119) -> Matrix(1,2,-2,-3) Matrix(73,192,46,121) -> Matrix(1,4,-2,-7) Matrix(37,96,-32,-83) -> Matrix(3,2,-2,-1) Matrix(11,24,-6,-13) -> Matrix(1,0,0,1) Matrix(83,144,34,59) -> Matrix(1,0,-2,1) Matrix(85,144,36,61) -> Matrix(1,6,-2,-11) Matrix(119,192,44,71) -> Matrix(1,2,-2,-3) Matrix(121,192,46,73) -> Matrix(1,4,-2,-7) Matrix(47,72,-32,-49) -> Matrix(3,8,-2,-5) Matrix(899,1296,274,395) -> Matrix(3,4,-10,-13) Matrix(901,1296,276,397) -> Matrix(3,2,-8,-5) Matrix(407,576,118,167) -> Matrix(1,2,-2,-3) Matrix(409,576,120,169) -> Matrix(1,2,-4,-7) Matrix(35,48,8,11) -> Matrix(1,2,-2,-3) Matrix(37,48,10,13) -> Matrix(1,2,-4,-7) Matrix(59,72,-50,-61) -> Matrix(1,0,0,1) Matrix(1,0,2,1) -> Matrix(3,4,-4,-5) Matrix(83,-96,32,-37) -> Matrix(3,2,-2,-1) Matrix(97,-120,38,-47) -> Matrix(3,2,-2,-1) Matrix(169,-240,50,-71) -> Matrix(3,2,-8,-5) Matrix(49,-72,32,-47) -> Matrix(11,8,-18,-13) Matrix(277,-432,84,-131) -> Matrix(3,2,-14,-9) Matrix(59,-96,8,-13) -> Matrix(3,2,-2,-1) Matrix(13,-24,6,-11) -> Matrix(1,0,0,1) Matrix(25,-72,8,-23) -> Matrix(3,2,-8,-5) 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): 7 Degree of the the map X: 7 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 0/1 -1/1 2 1 1/1 -1/1 1 12 6/5 0 2 5/4 (-3/4,-2/3) 0 12 4/3 -2/3 1 3 7/5 -1/1 1 12 10/7 0 6 3/2 -2/3 1 4 14/9 0 6 11/7 -3/5 1 12 8/5 -1/2 1 3 13/8 (-1/1,-2/3) 0 12 5/3 -3/5 1 12 2/1 0 6 7/3 -3/5 1 12 12/5 -1/2 3 1 5/2 (-1/2,0/1) 0 12 18/7 0 2 13/5 -1/1 1 12 8/3 -1/2 1 3 11/4 (-1/1,0/1) 0 12 3/1 -1/2 1 4 13/4 (-2/5,-1/3) 0 12 36/11 -1/3 3 1 23/7 -1/3 1 12 10/3 0 6 17/5 -1/3 1 12 24/7 0/1 1 1 7/2 (-1/2,0/1) 0 12 4/1 0/1 1 3 5/1 -1/1 1 12 6/1 0/1 2 2 7/1 1/1 1 12 1/0 (-1/1,0/1) 0 12 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(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(37,-48,10,-13) (5/4,4/3) -> (7/2,4/1) Glide Reflection Matrix(35,-48,8,-11) (4/3,7/5) -> (4/1,5/1) Glide Reflection 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(121,-192,46,-73) (11/7,8/5) -> (13/5,8/3) Glide Reflection Matrix(119,-192,44,-71) (8/5,13/8) -> (8/3,11/4) Glide Reflection 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(71,-168,30,-71) (7/3,12/5) -> (7/3,12/5) Reflection Matrix(49,-120,20,-49) (12/5,5/2) -> (12/5,5/2) Reflection Matrix(25,-72,8,-23) (11/4,3/1) -> (3/1,13/4) Parabolic Matrix(287,-936,88,-287) (13/4,36/11) -> (13/4,36/11) Reflection Matrix(505,-1656,154,-505) (36/11,23/7) -> (36/11,23/7) Reflection Matrix(239,-816,70,-239) (17/5,24/7) -> (17/5,24/7) Reflection Matrix(97,-336,28,-97) (24/7,7/2) -> (24/7,7/2) 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(5,4,-6,-5) (0/1,1/1) -> (-1/1,-2/3) Matrix(83,-96,32,-37) -> Matrix(3,2,-2,-1) -1/1 Matrix(97,-120,38,-47) -> Matrix(3,2,-2,-1) -1/1 Matrix(37,-48,10,-13) -> Matrix(3,2,-10,-7) Matrix(35,-48,8,-11) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(169,-240,50,-71) -> Matrix(3,2,-8,-5) -1/2 Matrix(49,-72,32,-47) -> Matrix(11,8,-18,-13) -2/3 Matrix(277,-432,84,-131) -> Matrix(3,2,-14,-9) Matrix(121,-192,46,-73) -> Matrix(7,4,-12,-7) *** -> (-2/3,-1/2) Matrix(119,-192,44,-71) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(59,-96,8,-13) -> Matrix(3,2,-2,-1) -1/1 Matrix(13,-24,6,-11) -> Matrix(1,0,0,1) Matrix(71,-168,30,-71) -> Matrix(11,6,-20,-11) (7/3,12/5) -> (-3/5,-1/2) Matrix(49,-120,20,-49) -> Matrix(-1,0,4,1) (12/5,5/2) -> (-1/2,0/1) Matrix(25,-72,8,-23) -> Matrix(3,2,-8,-5) -1/2 Matrix(287,-936,88,-287) -> Matrix(11,4,-30,-11) (13/4,36/11) -> (-2/5,-1/3) Matrix(505,-1656,154,-505) -> Matrix(7,2,-24,-7) (36/11,23/7) -> (-1/3,-1/4) Matrix(239,-816,70,-239) -> Matrix(-1,0,6,1) (17/5,24/7) -> (-1/3,0/1) Matrix(97,-336,28,-97) -> Matrix(-1,0,4,1) (24/7,7/2) -> (-1/2,0/1) Matrix(13,-72,2,-11) -> Matrix(1,0,2,1) 0/1 ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.