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 -4/1 -3/1 -2/1 -9/5 -3/2 -4/3 0/1 1/1 6/5 4/3 3/2 12/7 9/5 2/1 24/11 12/5 5/2 3/1 11/3 4/1 5/1 6/1 7/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -1/1 -5/1 -1/1 -1/2 -4/1 -2/3 0/1 -3/1 -1/2 1/0 -8/3 -2/3 0/1 -13/5 -1/1 -1/2 -5/2 -1/1 -1/2 -12/5 -1/2 -7/3 -1/2 -1/3 -2/1 0/1 -11/6 -1/1 0/1 -9/5 -1/2 1/0 -7/4 -1/1 1/0 -12/7 -1/1 -5/3 -1/1 -1/2 -8/5 -2/3 0/1 -11/7 -1/1 -1/2 -3/2 -1/2 -13/9 -1/2 -1/3 -36/25 -1/3 -23/16 -1/3 0/1 -10/7 0/1 -7/5 -1/2 -1/3 -18/13 -1/3 -11/8 -1/3 0/1 -4/3 0/1 -13/10 -1/1 0/1 -9/7 -1/2 1/0 -5/4 -1/1 -1/2 -11/9 -1/1 -1/2 -6/5 -1/2 -1/1 -1/2 0/1 0/1 0/1 1/1 0/1 1/0 6/5 1/0 5/4 -1/1 1/0 9/7 -1/2 1/0 4/3 0/1 15/11 1/2 1/0 11/8 0/1 1/1 7/5 1/1 1/0 3/2 1/0 11/7 -1/1 1/0 8/5 -2/1 0/1 5/3 -1/1 1/0 12/7 -1/1 7/4 -1/1 -1/2 9/5 -1/2 1/0 11/6 -1/1 0/1 2/1 0/1 13/6 0/1 1/1 24/11 1/1 11/5 1/1 1/0 9/4 1/0 7/3 1/1 1/0 12/5 1/0 5/2 -1/1 1/0 3/1 -1/2 1/0 7/2 -1/1 1/0 18/5 1/0 11/3 -1/1 1/0 4/1 -2/1 0/1 5/1 -1/1 1/0 6/1 -1/1 7/1 -1/1 -1/2 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(11,84,-8,-61) (-6/1,1/0) -> (-18/13,-11/8) Hyperbolic Matrix(11,60,2,11) (-6/1,-5/1) -> (5/1,6/1) Hyperbolic Matrix(13,60,8,37) (-5/1,-4/1) -> (8/5,5/3) Hyperbolic Matrix(11,36,-4,-13) (-4/1,-3/1) -> (-3/1,-8/3) Parabolic Matrix(73,192,46,121) (-8/3,-13/5) -> (11/7,8/5) Hyperbolic Matrix(47,120,-38,-97) (-13/5,-5/2) -> (-5/4,-11/9) Hyperbolic Matrix(49,120,20,49) (-5/2,-12/5) -> (12/5,5/2) Hyperbolic Matrix(71,168,30,71) (-12/5,-7/3) -> (7/3,12/5) Hyperbolic Matrix(37,84,-26,-59) (-7/3,-2/1) -> (-10/7,-7/5) Hyperbolic Matrix(83,156,-58,-109) (-2/1,-11/6) -> (-23/16,-10/7) Hyperbolic Matrix(145,264,106,193) (-11/6,-9/5) -> (15/11,11/8) Hyperbolic Matrix(61,108,48,85) (-9/5,-7/4) -> (5/4,9/7) Hyperbolic Matrix(97,168,56,97) (-7/4,-12/7) -> (12/7,7/4) Hyperbolic Matrix(71,120,42,71) (-12/7,-5/3) -> (5/3,12/7) Hyperbolic Matrix(37,60,8,13) (-5/3,-8/5) -> (4/1,5/1) Hyperbolic Matrix(83,132,22,35) (-8/5,-11/7) -> (11/3,4/1) Hyperbolic Matrix(47,72,-32,-49) (-11/7,-3/2) -> (-3/2,-13/9) Parabolic Matrix(491,708,224,323) (-13/9,-36/25) -> (24/11,11/5) Hyperbolic Matrix(709,1020,326,469) (-36/25,-23/16) -> (13/6,24/11) Hyperbolic Matrix(121,168,18,25) (-7/5,-18/13) -> (6/1,7/1) Hyperbolic Matrix(71,96,-54,-73) (-11/8,-4/3) -> (-4/3,-13/10) Parabolic 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(217,264,60,73) (-11/9,-6/5) -> (18/5,11/3) Hyperbolic Matrix(11,12,10,11) (-6/5,-1/1) -> (1/1,6/5) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(107,-132,30,-37) (6/5,5/4) -> (7/2,18/5) Hyperbolic Matrix(73,-96,54,-71) (9/7,4/3) -> (4/3,15/11) Parabolic Matrix(61,-84,8,-11) (11/8,7/5) -> (7/1,1/0) Hyperbolic Matrix(59,-84,26,-37) (7/5,3/2) -> (9/4,7/3) Hyperbolic Matrix(85,-132,38,-59) (3/2,11/7) -> (11/5,9/4) Hyperbolic Matrix(25,-48,12,-23) (11/6,2/1) -> (2/1,13/6) Parabolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(11,84,-8,-61) -> Matrix(1,0,-2,1) Matrix(11,60,2,11) -> Matrix(3,2,-2,-1) Matrix(13,60,8,37) -> Matrix(3,2,-2,-1) Matrix(11,36,-4,-13) -> Matrix(1,0,0,1) Matrix(73,192,46,121) -> Matrix(3,2,-2,-1) Matrix(47,120,-38,-97) -> Matrix(1,0,0,1) Matrix(49,120,20,49) -> Matrix(3,2,-2,-1) Matrix(71,168,30,71) -> Matrix(5,2,2,1) Matrix(37,84,-26,-59) -> Matrix(1,0,0,1) Matrix(83,156,-58,-109) -> Matrix(1,0,-2,1) Matrix(145,264,106,193) -> Matrix(1,0,2,1) Matrix(61,108,48,85) -> Matrix(1,0,0,1) Matrix(97,168,56,97) -> Matrix(1,2,-2,-3) Matrix(71,120,42,71) -> Matrix(3,2,-2,-1) Matrix(37,60,8,13) -> Matrix(3,2,-2,-1) Matrix(83,132,22,35) -> Matrix(3,2,-2,-1) Matrix(47,72,-32,-49) -> Matrix(3,2,-8,-5) Matrix(491,708,224,323) -> Matrix(5,2,2,1) Matrix(709,1020,326,469) -> Matrix(1,0,4,1) Matrix(121,168,18,25) -> Matrix(5,2,-8,-3) Matrix(71,96,-54,-73) -> Matrix(1,0,2,1) Matrix(167,216,92,119) -> Matrix(1,0,0,1) Matrix(85,108,48,61) -> Matrix(1,0,0,1) Matrix(217,264,60,73) -> Matrix(3,2,-2,-1) Matrix(11,12,10,11) -> Matrix(1,0,2,1) Matrix(1,0,2,1) -> Matrix(1,0,2,1) Matrix(107,-132,30,-37) -> Matrix(1,0,0,1) Matrix(73,-96,54,-71) -> Matrix(1,0,2,1) Matrix(61,-84,8,-11) -> Matrix(1,0,-2,1) Matrix(59,-84,26,-37) -> Matrix(1,0,0,1) Matrix(85,-132,38,-59) -> Matrix(1,2,0,1) Matrix(25,-48,12,-23) -> Matrix(1,0,2,1) Matrix(13,-36,4,-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): 5 Degree of the the map X: 5 Degree of the the map Y: 32 Permutation triple for Y: ((2,6,22,31,19,7)(3,11,28,29,12,4)(5,17,18)(8,24,13)(9,20,10)(14,30,15)(16,27)(25,26); (1,4,15,16,5,2)(3,10)(6,9,25,13,12,21)(7,8)(11,23,19,18,26,14)(17,29)(20,31,32,28,24,27)(22,30); (1,2,8,25,18,29,32,31,30,26,9,3)(4,13,28,14)(5,19,20,6)(7,23,11,10,27,15,22,21,12,17,16,24)) ----------------------------------------------------------------------- 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 2 2 1/1 (0/1,1/0) 0 12 6/5 1/0 1 2 5/4 (-1/1,1/0) 0 12 9/7 0 4 4/3 0/1 2 6 15/11 0 4 11/8 (0/1,1/1) 0 12 7/5 (1/1,1/0) 0 12 3/2 1/0 1 4 11/7 (-1/1,1/0) 0 12 8/5 0 6 5/3 (-1/1,1/0) 0 12 12/7 -1/1 4 2 7/4 (-1/1,-1/2) 0 12 9/5 0 4 11/6 (-1/1,0/1) 0 12 2/1 0/1 1 6 13/6 (0/1,1/1) 0 12 24/11 1/1 2 2 11/5 (1/1,1/0) 0 12 9/4 1/0 1 4 7/3 (1/1,1/0) 0 12 12/5 1/0 4 2 5/2 (-1/1,1/0) 0 12 3/1 0 4 7/2 (-1/1,1/0) 0 12 18/5 1/0 1 2 11/3 (-1/1,1/0) 0 12 4/1 0 6 5/1 (-1/1,1/0) 0 12 6/1 -1/1 2 2 7/1 (-1/1,-1/2) 0 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(11,-12,10,-11) (1/1,6/5) -> (1/1,6/5) Reflection Matrix(107,-132,30,-37) (6/5,5/4) -> (7/2,18/5) Hyperbolic Matrix(85,-108,48,-61) (5/4,9/7) -> (7/4,9/5) Glide Reflection Matrix(73,-96,54,-71) (9/7,4/3) -> (4/3,15/11) Parabolic Matrix(193,-264,106,-145) (15/11,11/8) -> (9/5,11/6) Glide Reflection Matrix(61,-84,8,-11) (11/8,7/5) -> (7/1,1/0) Hyperbolic Matrix(59,-84,26,-37) (7/5,3/2) -> (9/4,7/3) Hyperbolic Matrix(85,-132,38,-59) (3/2,11/7) -> (11/5,9/4) Hyperbolic Matrix(83,-132,22,-35) (11/7,8/5) -> (11/3,4/1) Glide Reflection Matrix(37,-60,8,-13) (8/5,5/3) -> (4/1,5/1) Glide Reflection Matrix(71,-120,42,-71) (5/3,12/7) -> (5/3,12/7) Reflection Matrix(97,-168,56,-97) (12/7,7/4) -> (12/7,7/4) Reflection Matrix(25,-48,12,-23) (11/6,2/1) -> (2/1,13/6) Parabolic Matrix(287,-624,132,-287) (13/6,24/11) -> (13/6,24/11) Reflection Matrix(241,-528,110,-241) (24/11,11/5) -> (24/11,11/5) Reflection 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(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic Matrix(109,-396,30,-109) (18/5,11/3) -> (18/5,11/3) Reflection Matrix(11,-60,2,-11) (5/1,6/1) -> (5/1,6/1) Reflection Matrix(13,-84,2,-13) (6/1,7/1) -> (6/1,7/1) 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(1,0,2,-1) -> Matrix(1,0,0,-1) (0/1,1/1) -> (0/1,1/0) Matrix(11,-12,10,-11) -> Matrix(1,0,0,-1) (1/1,6/5) -> (0/1,1/0) Matrix(107,-132,30,-37) -> Matrix(1,0,0,1) Matrix(85,-108,48,-61) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(73,-96,54,-71) -> Matrix(1,0,2,1) 0/1 Matrix(193,-264,106,-145) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(61,-84,8,-11) -> Matrix(1,0,-2,1) 0/1 Matrix(59,-84,26,-37) -> Matrix(1,0,0,1) Matrix(85,-132,38,-59) -> Matrix(1,2,0,1) 1/0 Matrix(83,-132,22,-35) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(37,-60,8,-13) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(71,-120,42,-71) -> Matrix(1,2,0,-1) (5/3,12/7) -> (-1/1,1/0) Matrix(97,-168,56,-97) -> Matrix(3,2,-4,-3) (12/7,7/4) -> (-1/1,-1/2) Matrix(25,-48,12,-23) -> Matrix(1,0,2,1) 0/1 Matrix(287,-624,132,-287) -> Matrix(1,0,2,-1) (13/6,24/11) -> (0/1,1/1) Matrix(241,-528,110,-241) -> Matrix(-1,2,0,1) (24/11,11/5) -> (1/1,1/0) Matrix(71,-168,30,-71) -> Matrix(-1,2,0,1) (7/3,12/5) -> (1/1,1/0) Matrix(49,-120,20,-49) -> Matrix(1,2,0,-1) (12/5,5/2) -> (-1/1,1/0) Matrix(13,-36,4,-11) -> Matrix(1,0,0,1) Matrix(109,-396,30,-109) -> Matrix(1,2,0,-1) (18/5,11/3) -> (-1/1,1/0) Matrix(11,-60,2,-11) -> Matrix(1,2,0,-1) (5/1,6/1) -> (-1/1,1/0) Matrix(13,-84,2,-13) -> Matrix(3,2,-4,-3) (6/1,7/1) -> (-1/1,-1/2) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.