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 -3/4 -2/3 -5/9 -1/2 -11/24 -1/3 -3/11 -1/4 -1/7 0/1 1/6 1/5 1/4 1/3 2/5 5/12 1/2 5/9 7/12 2/3 3/4 5/6 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 1/0 -5/6 0/1 -4/5 1/1 1/0 -7/9 -1/1 0/1 1/0 -3/4 0/1 -11/15 0/1 1/2 1/1 -8/11 0/1 1/1 -5/7 0/1 1/1 -2/3 1/0 -7/11 -1/1 0/1 -5/8 0/1 -3/5 1/1 1/0 -7/12 1/0 -4/7 -3/1 1/0 -5/9 -2/1 -1/1 1/0 -6/11 -2/1 -1/1 -1/2 -1/1 0/1 1/0 -6/13 -2/1 -1/1 -11/24 -1/1 -5/11 -1/1 0/1 -4/9 1/0 -3/7 -2/1 -1/1 -5/12 -1/1 -2/5 -1/1 -1/2 -1/3 -1/1 0/1 1/0 -2/7 -1/1 -1/2 -5/18 0/1 -3/11 -1/1 0/1 -1/4 0/1 -1/5 -1/1 1/0 -1/6 -1/1 -1/7 -1/1 0/1 0/1 -1/1 0/1 1/6 -1/1 1/5 -1/1 -1/2 1/4 0/1 1/3 -1/1 -1/2 0/1 3/8 0/1 5/13 -1/1 0/1 2/5 -1/1 1/0 5/12 -1/1 3/7 -1/1 -2/3 1/2 -1/1 -1/2 0/1 6/11 -1/1 -2/3 5/9 -1/1 -2/3 -1/2 4/7 -3/5 -1/2 7/12 -1/2 3/5 -1/2 -1/3 5/8 0/1 7/11 -1/1 0/1 2/3 -1/2 9/13 -2/5 -1/3 25/36 -1/3 16/23 -1/3 0/1 7/10 -1/2 -2/5 -1/3 5/7 -1/3 0/1 13/18 -1/3 8/11 -1/3 0/1 3/4 0/1 10/13 -1/1 0/1 7/9 -1/1 -1/2 0/1 4/5 -1/2 -1/3 9/11 -1/3 0/1 5/6 0/1 1/1 -1/2 0/1 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(11,10,12,11) (-1/1,-5/6) -> (5/6,1/1) Hyperbolic Matrix(37,30,-132,-107) (-5/6,-4/5) -> (-2/7,-5/18) Hyperbolic Matrix(61,48,108,85) (-4/5,-7/9) -> (5/9,4/7) Hyperbolic Matrix(71,54,-96,-73) (-7/9,-3/4) -> (-3/4,-11/15) Parabolic Matrix(145,106,264,193) (-11/15,-8/11) -> (6/11,5/9) Hyperbolic Matrix(11,8,-84,-61) (-8/11,-5/7) -> (-1/7,0/1) Hyperbolic Matrix(37,26,-84,-59) (-5/7,-2/3) -> (-4/9,-3/7) Hyperbolic Matrix(59,38,-132,-85) (-2/3,-7/11) -> (-5/11,-4/9) Hyperbolic Matrix(73,46,192,121) (-7/11,-5/8) -> (3/8,5/13) Hyperbolic Matrix(13,8,60,37) (-5/8,-3/5) -> (1/5,1/4) Hyperbolic Matrix(71,42,120,71) (-3/5,-7/12) -> (7/12,3/5) Hyperbolic Matrix(97,56,168,97) (-7/12,-4/7) -> (4/7,7/12) Hyperbolic Matrix(85,48,108,61) (-4/7,-5/9) -> (7/9,4/5) Hyperbolic Matrix(167,92,216,119) (-5/9,-6/11) -> (10/13,7/9) Hyperbolic Matrix(23,12,-48,-25) (-6/11,-1/2) -> (-1/2,-6/13) Parabolic Matrix(709,326,1020,469) (-6/13,-11/24) -> (25/36,16/23) Hyperbolic Matrix(491,224,708,323) (-11/24,-5/11) -> (9/13,25/36) Hyperbolic Matrix(71,30,168,71) (-3/7,-5/12) -> (5/12,3/7) Hyperbolic Matrix(49,20,120,49) (-5/12,-2/5) -> (2/5,5/12) Hyperbolic Matrix(11,4,-36,-13) (-2/5,-1/3) -> (-1/3,-2/7) Parabolic Matrix(217,60,264,73) (-5/18,-3/11) -> (9/11,5/6) Hyperbolic Matrix(83,22,132,35) (-3/11,-1/4) -> (5/8,7/11) Hyperbolic Matrix(37,8,60,13) (-1/4,-1/5) -> (3/5,5/8) Hyperbolic Matrix(11,2,60,11) (-1/5,-1/6) -> (1/6,1/5) Hyperbolic Matrix(121,18,168,25) (-1/6,-1/7) -> (5/7,13/18) Hyperbolic Matrix(61,-8,84,-11) (0/1,1/6) -> (13/18,8/11) Hyperbolic Matrix(13,-4,36,-11) (1/4,1/3) -> (1/3,3/8) Parabolic Matrix(97,-38,120,-47) (5/13,2/5) -> (4/5,9/11) Hyperbolic Matrix(59,-26,84,-37) (3/7,1/2) -> (7/10,5/7) Hyperbolic Matrix(109,-58,156,-83) (1/2,6/11) -> (16/23,7/10) Hyperbolic Matrix(49,-32,72,-47) (7/11,2/3) -> (2/3,9/13) Parabolic Matrix(73,-54,96,-71) (8/11,3/4) -> (3/4,10/13) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-2,1) Matrix(11,10,12,11) -> Matrix(1,0,-2,1) Matrix(37,30,-132,-107) -> Matrix(1,0,-2,1) Matrix(61,48,108,85) -> Matrix(1,2,-2,-3) Matrix(71,54,-96,-73) -> Matrix(1,0,2,1) Matrix(145,106,264,193) -> Matrix(3,-2,-4,3) Matrix(11,8,-84,-61) -> Matrix(1,0,-2,1) Matrix(37,26,-84,-59) -> Matrix(1,-2,0,1) Matrix(59,38,-132,-85) -> Matrix(1,0,0,1) Matrix(73,46,192,121) -> Matrix(1,0,0,1) Matrix(13,8,60,37) -> Matrix(1,0,-2,1) Matrix(71,42,120,71) -> Matrix(1,-2,-2,5) Matrix(97,56,168,97) -> Matrix(1,6,-2,-11) Matrix(85,48,108,61) -> Matrix(1,2,-2,-3) Matrix(167,92,216,119) -> Matrix(1,2,-2,-3) Matrix(23,12,-48,-25) -> Matrix(1,0,0,1) Matrix(709,326,1020,469) -> Matrix(1,2,-4,-7) Matrix(491,224,708,323) -> Matrix(3,2,-8,-5) Matrix(71,30,168,71) -> Matrix(3,4,-4,-5) Matrix(49,20,120,49) -> Matrix(3,2,-2,-1) Matrix(11,4,-36,-13) -> Matrix(1,0,0,1) Matrix(217,60,264,73) -> Matrix(1,0,-2,1) Matrix(83,22,132,35) -> Matrix(1,0,0,1) Matrix(37,8,60,13) -> Matrix(1,0,-2,1) Matrix(11,2,60,11) -> Matrix(1,2,-2,-3) Matrix(121,18,168,25) -> Matrix(1,0,-2,1) Matrix(61,-8,84,-11) -> Matrix(1,0,-2,1) Matrix(13,-4,36,-11) -> Matrix(1,0,0,1) Matrix(97,-38,120,-47) -> Matrix(1,0,-2,1) Matrix(59,-26,84,-37) -> Matrix(3,2,-8,-5) Matrix(109,-58,156,-83) -> Matrix(3,2,-8,-5) Matrix(49,-32,72,-47) -> Matrix(3,2,-8,-5) Matrix(73,-54,96,-71) -> 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): 6 Degree of the the map X: 6 Degree of the the map Y: 32 Permutation triple for Y: ((1,4,15,28,27,31,32,30,23,16,5,2)(3,10,29,11)(6,17,25,24,12,21,20,9,26,14,13,18)(7,22,19,8); (1,2,8,25,9,3)(4,14)(5,6)(7,21,12,11,16,15)(10,28,23,19,18,13)(17,29,32,31,22,26)(20,27)(24,30); (2,6,19,31,20,7)(3,12,30,29,13,4)(5,11,17)(8,23,24)(9,27,10)(14,22,15)(16,28)(25,26)) ----------------------------------------------------------------------- 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 (-1/1,0/1) 0 12 1/6 -1/1 1 2 1/5 (-1/1,-1/2) 0 12 1/4 0/1 1 3 1/3 0 4 3/8 0/1 1 3 5/13 (-1/1,0/1) 0 12 2/5 (-1/1,1/0) 0 12 5/12 -1/1 3 1 3/7 (-1/1,-2/3) 0 12 1/2 0 6 6/11 (-1/1,-2/3) 0 12 5/9 0 4 4/7 (-3/5,-1/2) 0 12 7/12 -1/2 4 1 3/5 (-1/2,-1/3) 0 12 5/8 0/1 1 3 7/11 (-1/1,0/1) 0 12 2/3 -1/2 1 4 9/13 (-2/5,-1/3) 0 12 25/36 -1/3 2 1 16/23 (-1/3,0/1) 0 12 7/10 0 6 5/7 (-1/3,0/1) 0 12 13/18 -1/3 1 2 8/11 (-1/3,0/1) 0 12 3/4 0/1 1 3 10/13 (-1/1,0/1) 0 12 7/9 0 4 4/5 (-1/2,-1/3) 0 12 9/11 (-1/3,0/1) 0 12 5/6 0/1 1 2 1/1 (-1/2,0/1) 0 12 1/0 0/1 1 1 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(61,-8,84,-11) (0/1,1/6) -> (13/18,8/11) Hyperbolic Matrix(11,-2,60,-11) (1/6,1/5) -> (1/6,1/5) Reflection Matrix(37,-8,60,-13) (1/5,1/4) -> (3/5,5/8) Glide Reflection Matrix(13,-4,36,-11) (1/4,1/3) -> (1/3,3/8) Parabolic Matrix(121,-46,192,-73) (3/8,5/13) -> (5/8,7/11) Glide Reflection Matrix(97,-38,120,-47) (5/13,2/5) -> (4/5,9/11) Hyperbolic Matrix(49,-20,120,-49) (2/5,5/12) -> (2/5,5/12) Reflection Matrix(71,-30,168,-71) (5/12,3/7) -> (5/12,3/7) Reflection Matrix(59,-26,84,-37) (3/7,1/2) -> (7/10,5/7) Hyperbolic Matrix(109,-58,156,-83) (1/2,6/11) -> (16/23,7/10) Hyperbolic Matrix(167,-92,216,-119) (6/11,5/9) -> (10/13,7/9) Glide Reflection Matrix(85,-48,108,-61) (5/9,4/7) -> (7/9,4/5) Glide Reflection Matrix(97,-56,168,-97) (4/7,7/12) -> (4/7,7/12) Reflection Matrix(71,-42,120,-71) (7/12,3/5) -> (7/12,3/5) Reflection Matrix(49,-32,72,-47) (7/11,2/3) -> (2/3,9/13) Parabolic Matrix(649,-450,936,-649) (9/13,25/36) -> (9/13,25/36) Reflection Matrix(1151,-800,1656,-1151) (25/36,16/23) -> (25/36,16/23) Reflection Matrix(181,-130,252,-181) (5/7,13/18) -> (5/7,13/18) Reflection Matrix(73,-54,96,-71) (8/11,3/4) -> (3/4,10/13) Parabolic Matrix(109,-90,132,-109) (9/11,5/6) -> (9/11,5/6) Reflection Matrix(11,-10,12,-11) (5/6,1/1) -> (5/6,1/1) Reflection Matrix(-1,2,0,1) (1/1,1/0) -> (1/1,1/0) 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(61,-8,84,-11) -> Matrix(1,0,-2,1) 0/1 Matrix(11,-2,60,-11) -> Matrix(3,2,-4,-3) (1/6,1/5) -> (-1/1,-1/2) Matrix(37,-8,60,-13) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(13,-4,36,-11) -> Matrix(1,0,0,1) Matrix(121,-46,192,-73) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(97,-38,120,-47) -> Matrix(1,0,-2,1) 0/1 Matrix(49,-20,120,-49) -> Matrix(1,2,0,-1) (2/5,5/12) -> (-1/1,1/0) Matrix(71,-30,168,-71) -> Matrix(5,4,-6,-5) (5/12,3/7) -> (-1/1,-2/3) Matrix(59,-26,84,-37) -> Matrix(3,2,-8,-5) -1/2 Matrix(109,-58,156,-83) -> Matrix(3,2,-8,-5) -1/2 Matrix(167,-92,216,-119) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(85,-48,108,-61) -> Matrix(3,2,-4,-3) *** -> (-1/1,-1/2) Matrix(97,-56,168,-97) -> Matrix(11,6,-20,-11) (4/7,7/12) -> (-3/5,-1/2) Matrix(71,-42,120,-71) -> Matrix(5,2,-12,-5) (7/12,3/5) -> (-1/2,-1/3) Matrix(49,-32,72,-47) -> Matrix(3,2,-8,-5) -1/2 Matrix(649,-450,936,-649) -> Matrix(11,4,-30,-11) (9/13,25/36) -> (-2/5,-1/3) Matrix(1151,-800,1656,-1151) -> Matrix(-1,0,6,1) (25/36,16/23) -> (-1/3,0/1) Matrix(181,-130,252,-181) -> Matrix(-1,0,6,1) (5/7,13/18) -> (-1/3,0/1) Matrix(73,-54,96,-71) -> Matrix(1,0,2,1) 0/1 Matrix(109,-90,132,-109) -> Matrix(-1,0,6,1) (9/11,5/6) -> (-1/3,0/1) Matrix(11,-10,12,-11) -> Matrix(-1,0,4,1) (5/6,1/1) -> (-1/2,0/1) Matrix(-1,2,0,1) -> Matrix(-1,0,4,1) (1/1,1/0) -> (-1/2,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.