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 -5/3 -3/2 -11/9 -1/1 -5/6 -2/3 -1/2 -1/3 -1/5 -1/6 0/1 1/6 1/5 1/3 3/7 1/2 5/9 7/11 2/3 1/1 3/2 5/3 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 0/1 1/2 1/1 -5/3 1/2 -8/5 1/2 3/5 2/3 -11/7 2/3 3/4 -3/2 1/1 -4/3 0/1 -5/4 1/3 1/1 -11/9 1/2 -6/5 1/2 2/3 1/1 -1/1 0/1 1/1 -6/7 1/2 2/3 1/1 -5/6 1/1 -9/11 1/1 2/1 -4/5 1/1 2/1 1/0 -7/9 1/0 -3/4 -1/1 1/1 -5/7 0/1 1/0 -7/10 -1/1 1/1 -2/3 0/1 -5/8 1/3 -3/5 0/1 1/2 -7/12 1/2 -4/7 1/2 3/5 2/3 -1/2 1/1 -1/3 1/0 -1/4 -1/1 -1/5 -1/2 0/1 -2/11 -1/4 -1/5 0/1 -1/6 0/1 0/1 0/1 1/1 1/0 1/6 1/0 1/5 -2/1 1/0 1/4 -1/1 1/3 -1/2 1/0 3/8 -1/1 5/13 -1/1 0/1 7/18 -1/1 2/5 -1/1 -1/2 0/1 5/12 0/1 3/7 0/1 1/0 1/2 -1/1 1/1 5/9 1/0 4/7 -1/1 0/1 1/0 7/12 1/0 3/5 -2/1 1/0 5/8 -1/1 7/11 -2/1 -1/1 2/3 -1/1 1/1 -1/1 0/1 4/3 -1/1 15/11 -1/1 -2/3 11/8 -1/1 7/5 -2/3 -1/2 17/12 -1/2 10/7 -1/1 -1/2 0/1 13/9 -1/2 3/2 -1/1 -1/3 11/7 -1/2 0/1 19/12 0/1 8/5 -1/1 0/1 1/0 5/3 -1/2 1/0 12/7 -1/1 0/1 1/0 19/11 -1/1 0/1 7/4 -1/1 2/1 -1/2 -1/3 0/1 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,4,0,1) (-2/1,1/0) -> (2/1,1/0) Parabolic Matrix(29,50,-18,-31) (-2/1,-5/3) -> (-5/3,-8/5) Parabolic Matrix(19,30,-102,-161) (-8/5,-11/7) -> (-1/5,-2/11) Hyperbolic Matrix(91,142,66,103) (-11/7,-3/2) -> (11/8,7/5) Hyperbolic Matrix(19,26,-30,-41) (-3/2,-4/3) -> (-2/3,-5/8) Hyperbolic Matrix(29,38,-42,-55) (-4/3,-5/4) -> (-7/10,-2/3) Hyperbolic Matrix(79,98,54,67) (-5/4,-11/9) -> (13/9,3/2) Hyperbolic Matrix(155,188,108,131) (-11/9,-6/5) -> (10/7,13/9) Hyperbolic Matrix(11,12,-12,-13) (-6/5,-1/1) -> (-1/1,-6/7) Parabolic Matrix(61,52,156,133) (-6/7,-5/6) -> (7/18,2/5) Hyperbolic Matrix(107,88,276,227) (-5/6,-9/11) -> (5/13,7/18) Hyperbolic Matrix(227,184,132,107) (-9/11,-4/5) -> (12/7,19/11) Hyperbolic Matrix(61,48,108,85) (-4/5,-7/9) -> (5/9,4/7) Hyperbolic Matrix(29,22,54,41) (-7/9,-3/4) -> (1/2,5/9) Hyperbolic Matrix(19,14,42,31) (-3/4,-5/7) -> (3/7,1/2) Hyperbolic Matrix(131,92,84,59) (-5/7,-7/10) -> (3/2,11/7) Hyperbolic Matrix(13,8,60,37) (-5/8,-3/5) -> (1/5,1/4) Hyperbolic Matrix(17,10,90,53) (-3/5,-7/12) -> (1/6,1/5) Hyperbolic Matrix(97,56,168,97) (-7/12,-4/7) -> (4/7,7/12) Hyperbolic Matrix(53,30,30,17) (-4/7,-1/2) -> (7/4,2/1) Hyperbolic Matrix(5,2,-18,-7) (-1/2,-1/3) -> (-1/3,-1/4) Parabolic Matrix(37,8,60,13) (-1/4,-1/5) -> (3/5,5/8) Hyperbolic Matrix(257,46,162,29) (-2/11,-1/6) -> (19/12,8/5) Hyperbolic Matrix(17,2,42,5) (-1/6,0/1) -> (2/5,5/12) Hyperbolic Matrix(77,-10,54,-7) (0/1,1/6) -> (17/12,10/7) Hyperbolic Matrix(13,-4,36,-11) (1/4,1/3) -> (1/3,3/8) Parabolic Matrix(263,-100,192,-73) (3/8,5/13) -> (15/11,11/8) Hyperbolic Matrix(265,-112,168,-71) (5/12,3/7) -> (11/7,19/12) Hyperbolic Matrix(169,-100,120,-71) (7/12,3/5) -> (7/5,17/12) Hyperbolic Matrix(229,-144,132,-83) (5/8,7/11) -> (19/11,7/4) Hyperbolic Matrix(89,-58,66,-43) (7/11,2/3) -> (4/3,15/11) Hyperbolic Matrix(7,-6,6,-5) (2/3,1/1) -> (1/1,4/3) Parabolic Matrix(61,-100,36,-59) (8/5,5/3) -> (5/3,12/7) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,0,-4,1) Matrix(29,50,-18,-31) -> Matrix(5,-2,8,-3) Matrix(19,30,-102,-161) -> Matrix(3,-2,-10,7) Matrix(91,142,66,103) -> Matrix(5,-4,-6,5) Matrix(19,26,-30,-41) -> Matrix(1,0,2,1) Matrix(29,38,-42,-55) -> Matrix(1,0,-2,1) Matrix(79,98,54,67) -> Matrix(1,0,-4,1) Matrix(155,188,108,131) -> Matrix(3,-2,-4,3) Matrix(11,12,-12,-13) -> Matrix(1,0,0,1) Matrix(61,52,156,133) -> Matrix(3,-2,-4,3) Matrix(107,88,276,227) -> Matrix(1,-2,0,1) Matrix(227,184,132,107) -> Matrix(1,-2,0,1) Matrix(61,48,108,85) -> Matrix(1,-2,0,1) Matrix(29,22,54,41) -> Matrix(1,0,0,1) Matrix(19,14,42,31) -> Matrix(1,0,0,1) Matrix(131,92,84,59) -> Matrix(1,0,-2,1) Matrix(13,8,60,37) -> Matrix(5,-2,-2,1) Matrix(17,10,90,53) -> Matrix(5,-2,-2,1) Matrix(97,56,168,97) -> Matrix(3,-2,2,-1) Matrix(53,30,30,17) -> Matrix(3,-2,-4,3) Matrix(5,2,-18,-7) -> Matrix(1,-2,0,1) Matrix(37,8,60,13) -> Matrix(3,2,-2,-1) Matrix(257,46,162,29) -> Matrix(1,0,4,1) Matrix(17,2,42,5) -> Matrix(1,0,-2,1) Matrix(77,-10,54,-7) -> Matrix(1,0,-2,1) Matrix(13,-4,36,-11) -> Matrix(1,0,0,1) Matrix(263,-100,192,-73) -> Matrix(1,2,-2,-3) Matrix(265,-112,168,-71) -> Matrix(1,0,-2,1) Matrix(169,-100,120,-71) -> Matrix(1,4,-2,-7) Matrix(229,-144,132,-83) -> Matrix(1,2,-2,-3) Matrix(89,-58,66,-43) -> Matrix(3,4,-4,-5) Matrix(7,-6,6,-5) -> Matrix(1,0,0,1) Matrix(61,-100,36,-59) -> 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: ((1,6,19,30,29,25,32,18,17,20,7,2)(3,11,26,28,13,27,24,23,16,21,12,4)(5,14,31,15)(8,10,22,9); (1,4,5)(2,9,24,25,10,3)(6,18)(8,21,19)(12,13)(14,30,16,15,20,26)(17,22,28)(27,31,32); (1,3)(2,7,15,27,12,8)(4,13,22,25,29,14)(5,16,23,9,17,6)(10,19,18,31,26,11)(20,28)(21,30)(24,32)) ----------------------------------------------------------------------- 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 -1/1 (0/1,1/1) 0 12 -5/6 1/1 2 2 -4/5 0 12 -7/9 1/0 4 4 -3/4 0 6 -5/7 (0/1,1/0) 0 12 -2/3 0/1 2 4 -3/5 (0/1,1/2) 0 12 -1/2 1/1 2 6 -1/3 1/0 4 4 -1/4 -1/1 2 6 -1/5 (-1/2,0/1) 0 12 -1/6 0/1 4 2 0/1 0 12 1/6 1/0 4 2 1/5 (-2/1,1/0) 0 12 1/4 -1/1 2 6 1/3 0 4 3/8 -1/1 2 6 5/13 (-1/1,0/1) 0 12 7/18 -1/1 2 2 2/5 0 12 5/12 0/1 4 2 3/7 (0/1,1/0) 0 12 1/2 0 6 5/9 1/0 4 4 4/7 0 12 7/12 1/0 4 2 3/5 (-2/1,1/0) 0 12 5/8 -1/1 2 6 7/11 (-2/1,-1/1) 0 12 2/3 -1/1 2 4 1/1 (-1/1,0/1) 0 12 1/0 0/1 2 2 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,-1) (-1/1,1/0) -> (-1/1,1/0) Reflection Matrix(11,10,-12,-11) (-1/1,-5/6) -> (-1/1,-5/6) Reflection Matrix(47,38,120,97) (-5/6,-4/5) -> (7/18,2/5) Glide Reflection Matrix(61,48,108,85) (-4/5,-7/9) -> (5/9,4/7) Hyperbolic Matrix(29,22,54,41) (-7/9,-3/4) -> (1/2,5/9) Hyperbolic Matrix(19,14,42,31) (-3/4,-5/7) -> (3/7,1/2) Hyperbolic Matrix(29,20,-42,-29) (-5/7,-2/3) -> (-5/7,-2/3) Reflection Matrix(19,12,-30,-19) (-2/3,-3/5) -> (-2/3,-3/5) Reflection Matrix(7,4,30,17) (-3/5,-1/2) -> (1/5,1/4) Glide Reflection Matrix(5,2,-18,-7) (-1/2,-1/3) -> (-1/3,-1/4) Parabolic Matrix(37,8,60,13) (-1/4,-1/5) -> (3/5,5/8) Hyperbolic Matrix(11,2,-60,-11) (-1/5,-1/6) -> (-1/5,-1/6) Reflection Matrix(17,2,42,5) (-1/6,0/1) -> (2/5,5/12) Hyperbolic Matrix(31,-4,54,-7) (0/1,1/6) -> (4/7,7/12) Glide Reflection Matrix(11,-2,60,-11) (1/6,1/5) -> (1/6,1/5) 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(181,-70,468,-181) (5/13,7/18) -> (5/13,7/18) Reflection Matrix(71,-30,168,-71) (5/12,3/7) -> (5/12,3/7) Reflection Matrix(71,-42,120,-71) (7/12,3/5) -> (7/12,3/5) Reflection Matrix(43,-28,66,-43) (7/11,2/3) -> (7/11,2/3) Reflection Matrix(5,-4,6,-5) (2/3,1/1) -> (2/3,1/1) 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(1,2,0,-1) -> Matrix(1,0,2,-1) (-1/1,1/0) -> (0/1,1/1) Matrix(11,10,-12,-11) -> Matrix(1,0,2,-1) (-1/1,-5/6) -> (0/1,1/1) Matrix(47,38,120,97) -> Matrix(1,-2,-2,3) Matrix(61,48,108,85) -> Matrix(1,-2,0,1) 1/0 Matrix(29,22,54,41) -> Matrix(1,0,0,1) Matrix(19,14,42,31) -> Matrix(1,0,0,1) Matrix(29,20,-42,-29) -> Matrix(1,0,0,-1) (-5/7,-2/3) -> (0/1,1/0) Matrix(19,12,-30,-19) -> Matrix(1,0,4,-1) (-2/3,-3/5) -> (0/1,1/2) Matrix(7,4,30,17) -> Matrix(3,-2,-2,1) Matrix(5,2,-18,-7) -> Matrix(1,-2,0,1) 1/0 Matrix(37,8,60,13) -> Matrix(3,2,-2,-1) -1/1 Matrix(11,2,-60,-11) -> Matrix(-1,0,4,1) (-1/5,-1/6) -> (-1/2,0/1) Matrix(17,2,42,5) -> Matrix(1,0,-2,1) 0/1 Matrix(31,-4,54,-7) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(11,-2,60,-11) -> Matrix(1,4,0,-1) (1/6,1/5) -> (-2/1,1/0) Matrix(13,-4,36,-11) -> Matrix(1,0,0,1) Matrix(121,-46,192,-73) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(181,-70,468,-181) -> Matrix(-1,0,2,1) (5/13,7/18) -> (-1/1,0/1) Matrix(71,-30,168,-71) -> Matrix(1,0,0,-1) (5/12,3/7) -> (0/1,1/0) Matrix(71,-42,120,-71) -> Matrix(1,4,0,-1) (7/12,3/5) -> (-2/1,1/0) Matrix(43,-28,66,-43) -> Matrix(3,4,-2,-3) (7/11,2/3) -> (-2/1,-1/1) Matrix(5,-4,6,-5) -> Matrix(-1,0,2,1) (2/3,1/1) -> (-1/1,0/1) Matrix(-1,2,0,1) -> Matrix(-1,0,2,1) (1/1,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.