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 -1/1 -2/3 -3/5 -5/9 -1/2 -3/7 -1/3 -3/11 -1/5 -1/6 0/1 1/3 1/2 2/3 5/6 1/1 11/9 3/2 5/3 11/6 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 -1/1 1/1 -7/4 1/0 -5/3 0/1 -13/8 1/2 -8/5 1/1 3/1 -11/7 -1/1 1/0 -3/2 0/1 -1/1 0/1 1/0 -3/4 0/1 -11/15 1/0 -8/11 -1/1 -1/3 -5/7 1/1 -7/10 1/0 -2/3 0/1 -7/11 0/1 1/1 -12/19 1/1 7/5 -5/8 1/0 -3/5 -1/1 -7/12 0/1 -4/7 1/3 1/1 -5/9 1/0 -1/2 0/1 -3/7 1/1 1/0 -8/19 3/1 5/1 -5/12 1/0 -2/5 -3/1 -1/1 -1/3 0/1 -2/7 3/5 1/1 -5/18 1/1 -3/11 1/1 2/1 -4/15 2/1 -1/4 1/0 -1/5 -1/1 0/1 -1/6 0/1 -1/7 1/3 0/1 -1/1 1/1 1/3 1/0 2/5 -3/1 -1/1 5/12 1/0 3/7 -3/1 1/2 1/0 4/7 -3/1 -1/1 7/12 -2/1 3/5 -2/1 -1/1 2/3 -1/1 5/7 -1/1 1/0 3/4 -2/1 7/9 -3/2 4/5 -9/7 -1/1 9/11 -8/7 -1/1 5/6 -1/1 1/1 -1/1 7/6 -1/1 6/5 -1/1 -9/11 11/9 -3/4 5/4 -2/3 9/7 -1/1 -1/2 4/3 -1/1 7/5 -1/1 -2/3 3/2 -1/2 5/3 -1/2 7/4 -1/2 9/5 -1/1 11/6 -1/2 2/1 -1/1 -1/3 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(17,30,30,53) (-2/1,-7/4) -> (1/2,4/7) Hyperbolic Matrix(59,100,-36,-61) (-7/4,-5/3) -> (-5/3,-13/8) Parabolic Matrix(121,196,-192,-311) (-13/8,-8/5) -> (-12/19,-5/8) Hyperbolic Matrix(71,112,-168,-265) (-8/5,-11/7) -> (-3/7,-8/19) Hyperbolic Matrix(53,82,42,65) (-11/7,-3/2) -> (5/4,9/7) Hyperbolic Matrix(5,6,-6,-7) (-3/2,-1/1) -> (-1/1,-3/4) Parabolic Matrix(119,88,96,71) (-3/4,-11/15) -> (11/9,5/4) Hyperbolic Matrix(211,154,174,127) (-11/15,-8/11) -> (6/5,11/9) Hyperbolic Matrix(11,8,-84,-61) (-8/11,-5/7) -> (-1/7,0/1) Hyperbolic Matrix(139,98,78,55) (-5/7,-7/10) -> (7/4,9/5) Hyperbolic Matrix(43,30,-162,-113) (-7/10,-2/3) -> (-4/15,-1/4) Hyperbolic Matrix(53,34,-198,-127) (-2/3,-7/11) -> (-3/11,-4/15) Hyperbolic Matrix(215,136,264,167) (-7/11,-12/19) -> (4/5,9/11) Hyperbolic Matrix(29,18,66,41) (-5/8,-3/5) -> (3/7,1/2) Hyperbolic Matrix(17,10,-114,-67) (-3/5,-7/12) -> (-1/6,-1/7) 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(41,22,54,29) (-5/9,-1/2) -> (3/4,7/9) Hyperbolic Matrix(31,14,42,19) (-1/2,-3/7) -> (5/7,3/4) Hyperbolic Matrix(233,98,126,53) (-8/19,-5/12) -> (11/6,2/1) 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(157,44,132,37) (-2/7,-5/18) -> (7/6,6/5) Hyperbolic Matrix(217,60,264,73) (-5/18,-3/11) -> (9/11,5/6) Hyperbolic Matrix(43,10,30,7) (-1/4,-1/5) -> (7/5,3/2) Hyperbolic Matrix(53,10,90,17) (-1/5,-1/6) -> (7/12,3/5) Hyperbolic Matrix(7,-2,18,-5) (0/1,1/3) -> (1/3,2/5) Parabolic Matrix(185,-78,102,-43) (5/12,3/7) -> (9/5,11/6) Hyperbolic Matrix(41,-26,30,-19) (3/5,2/3) -> (4/3,7/5) Hyperbolic Matrix(55,-38,42,-29) (2/3,5/7) -> (9/7,4/3) Hyperbolic Matrix(13,-12,12,-11) (5/6,1/1) -> (1/1,7/6) Parabolic Matrix(31,-50,18,-29) (3/2,5/3) -> (5/3,7/4) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,0,-2,1) Matrix(17,30,30,53) -> Matrix(1,-2,0,1) Matrix(59,100,-36,-61) -> Matrix(1,0,2,1) Matrix(121,196,-192,-311) -> Matrix(3,-2,2,-1) Matrix(71,112,-168,-265) -> Matrix(1,2,0,1) Matrix(53,82,42,65) -> Matrix(1,2,-2,-3) Matrix(5,6,-6,-7) -> Matrix(1,0,0,1) Matrix(119,88,96,71) -> Matrix(3,-2,-4,3) Matrix(211,154,174,127) -> Matrix(3,4,-4,-5) Matrix(11,8,-84,-61) -> Matrix(1,0,2,1) Matrix(139,98,78,55) -> Matrix(1,0,-2,1) Matrix(43,30,-162,-113) -> Matrix(1,2,0,1) Matrix(53,34,-198,-127) -> Matrix(3,-2,2,-1) Matrix(215,136,264,167) -> Matrix(7,-8,-6,7) Matrix(29,18,66,41) -> Matrix(1,-2,0,1) Matrix(17,10,-114,-67) -> Matrix(1,0,4,1) Matrix(97,56,168,97) -> Matrix(5,-2,-2,1) Matrix(85,48,108,61) -> Matrix(3,-4,-2,3) Matrix(41,22,54,29) -> Matrix(3,2,-2,-1) Matrix(31,14,42,19) -> Matrix(1,-2,0,1) Matrix(233,98,126,53) -> Matrix(1,-4,-2,9) Matrix(49,20,120,49) -> Matrix(1,0,0,1) Matrix(11,4,-36,-13) -> Matrix(1,0,2,1) Matrix(157,44,132,37) -> Matrix(7,-6,-8,7) Matrix(217,60,264,73) -> Matrix(9,-10,-8,9) Matrix(43,10,30,7) -> Matrix(1,2,-2,-3) Matrix(53,10,90,17) -> Matrix(3,2,-2,-1) Matrix(7,-2,18,-5) -> Matrix(1,-2,0,1) Matrix(185,-78,102,-43) -> Matrix(1,4,-2,-7) Matrix(41,-26,30,-19) -> Matrix(3,4,-4,-5) Matrix(55,-38,42,-29) -> Matrix(1,2,-2,-3) Matrix(13,-12,12,-11) -> Matrix(1,2,-2,-3) Matrix(31,-50,18,-29) -> Matrix(3,2,-8,-5) 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): 10 Degree of the the map X: 10 Degree of the the map Y: 32 Permutation triple for Y: ((1,6,18,21,8,20,31,30,26,19,7,2)(3,11,17,23,22,15,32,28,27,29,12,4)(5,16,14,13)(9,24,25,10); (1,4,14,31,15,5)(2,10,3)(7,8)(9,23,26,25,29,18)(11,28)(13,19,17)(16,21,27)(20,24,32); (1,3)(2,8,16,15,22,9)(4,12,25,20,7,13)(5,17,28,24,18,6)(10,26,30,14,27,11)(19,23)(21,29)(31,32)) ----------------------------------------------------------------------- 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 (0/1,1/0) 0 12 -2/3 0/1 1 4 -7/11 (0/1,1/1) 0 12 -5/8 1/0 1 6 -3/5 -1/1 2 12 -7/12 0/1 4 2 -4/7 0 12 -5/9 1/0 4 4 -1/2 0/1 1 6 -3/7 (1/1,1/0) 0 12 -5/12 1/0 4 2 -2/5 0 12 -1/3 0/1 2 4 -2/7 0 12 -3/11 (1/1,2/1) 0 12 -1/4 1/0 1 6 0/1 0 12 1/3 1/0 4 4 2/5 0 12 5/12 1/0 4 2 3/7 -3/1 2 12 1/2 1/0 1 6 4/7 0 12 7/12 -2/1 4 2 3/5 (-2/1,-1/1) 0 12 2/3 -1/1 1 4 5/7 (-1/1,1/0) 0 12 3/4 -2/1 1 6 7/9 -3/2 4 4 4/5 0 12 9/11 (-8/7,-1/1) 0 12 5/6 -1/1 7 2 1/1 -1/1 2 12 1/0 0/1 1 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(5,4,-6,-5) (-1/1,-2/3) -> (-1/1,-2/3) Reflection Matrix(43,28,-66,-43) (-2/3,-7/11) -> (-2/3,-7/11) Reflection Matrix(35,22,-132,-83) (-7/11,-5/8) -> (-3/11,-1/4) Glide Reflection Matrix(29,18,66,41) (-5/8,-3/5) -> (3/7,1/2) Hyperbolic Matrix(71,42,-120,-71) (-3/5,-7/12) -> (-3/5,-7/12) Reflection 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(41,22,54,29) (-5/9,-1/2) -> (3/4,7/9) Hyperbolic Matrix(31,14,42,19) (-1/2,-3/7) -> (5/7,3/4) Hyperbolic Matrix(71,30,-168,-71) (-3/7,-5/12) -> (-3/7,-5/12) Reflection 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(107,30,132,37) (-2/7,-3/11) -> (4/5,9/11) Glide Reflection Matrix(17,4,30,7) (-1/4,0/1) -> (1/2,4/7) Glide Reflection Matrix(7,-2,18,-5) (0/1,1/3) -> (1/3,2/5) Parabolic 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(19,-12,30,-19) (3/5,2/3) -> (3/5,2/3) Reflection Matrix(29,-20,42,-29) (2/3,5/7) -> (2/3,5/7) Reflection 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(1,2,0,-1) -> Matrix(1,0,0,-1) (-1/1,1/0) -> (0/1,1/0) Matrix(5,4,-6,-5) -> Matrix(1,0,0,-1) (-1/1,-2/3) -> (0/1,1/0) Matrix(43,28,-66,-43) -> Matrix(1,0,2,-1) (-2/3,-7/11) -> (0/1,1/1) Matrix(35,22,-132,-83) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(29,18,66,41) -> Matrix(1,-2,0,1) 1/0 Matrix(71,42,-120,-71) -> Matrix(-1,0,2,1) (-3/5,-7/12) -> (-1/1,0/1) Matrix(97,56,168,97) -> Matrix(5,-2,-2,1) Matrix(85,48,108,61) -> Matrix(3,-4,-2,3) Matrix(41,22,54,29) -> Matrix(3,2,-2,-1) -1/1 Matrix(31,14,42,19) -> Matrix(1,-2,0,1) 1/0 Matrix(71,30,-168,-71) -> Matrix(-1,2,0,1) (-3/7,-5/12) -> (1/1,1/0) Matrix(49,20,120,49) -> Matrix(1,0,0,1) Matrix(11,4,-36,-13) -> Matrix(1,0,2,1) 0/1 Matrix(107,30,132,37) -> Matrix(7,-6,-6,5) Matrix(17,4,30,7) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(7,-2,18,-5) -> Matrix(1,-2,0,1) 1/0 Matrix(71,-30,168,-71) -> Matrix(1,6,0,-1) (5/12,3/7) -> (-3/1,1/0) Matrix(71,-42,120,-71) -> Matrix(3,4,-2,-3) (7/12,3/5) -> (-2/1,-1/1) Matrix(19,-12,30,-19) -> Matrix(3,4,-2,-3) (3/5,2/3) -> (-2/1,-1/1) Matrix(29,-20,42,-29) -> Matrix(1,2,0,-1) (2/3,5/7) -> (-1/1,1/0) Matrix(109,-90,132,-109) -> Matrix(15,16,-14,-15) (9/11,5/6) -> (-8/7,-1/1) Matrix(11,-10,12,-11) -> Matrix(1,2,0,-1) (5/6,1/1) -> (-1/1,1/0) 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.