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): 216 Minimal number of generators: 37 Number of equivalence classes of cusps: 18 Genus: 10 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -6/1 -9/2 -3/1 -9/5 -3/2 0/1 1/1 3/2 27/17 9/5 2/1 18/7 3/1 27/8 18/5 9/2 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 1/1 -5/1 2/1 -9/2 1/0 -4/1 -1/1 -7/2 -1/2 -3/1 0/1 -11/4 1/0 -19/7 0/1 -27/10 1/2 1/0 -8/3 1/1 -5/2 1/0 -17/7 2/1 -12/5 1/1 -7/3 2/1 -9/4 1/0 -2/1 -1/1 -9/5 0/1 -7/4 1/2 -12/7 1/1 -5/3 0/1 -18/11 1/1 -13/8 3/2 -8/5 1/1 -3/2 1/0 -10/7 -1/1 -27/19 -2/1 0/1 -17/12 1/0 -7/5 -2/1 -18/13 -1/1 -11/8 -1/2 -4/3 -1/1 -9/7 0/1 -5/4 1/2 -11/9 0/1 -6/5 1/1 -1/1 0/1 0/1 -1/1 1/1 1/1 0/1 6/5 -1/1 5/4 -1/2 9/7 0/1 4/3 1/1 7/5 2/1 3/2 1/0 11/7 0/1 19/12 1/0 27/17 -2/1 0/1 8/5 -1/1 5/3 0/1 17/10 -1/2 12/7 -1/1 7/4 -1/2 9/5 0/1 2/1 1/1 9/4 1/0 7/3 -2/1 12/5 -1/1 5/2 1/0 18/7 -1/1 13/5 -2/3 8/3 -1/1 3/1 0/1 10/3 1/1 27/8 1/2 1/0 17/5 0/1 7/2 1/2 18/5 1/1 11/3 2/1 4/1 1/1 9/2 1/0 5/1 -2/1 11/2 1/0 6/1 -1/1 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(7,54,4,31) (-6/1,1/0) -> (12/7,7/4) Hyperbolic Matrix(29,162,-12,-67) (-6/1,-5/1) -> (-17/7,-12/5) Hyperbolic Matrix(23,108,10,47) (-5/1,-9/2) -> (9/4,7/3) Hyperbolic Matrix(13,54,6,25) (-9/2,-4/1) -> (2/1,9/4) Hyperbolic Matrix(29,108,-18,-67) (-4/1,-7/2) -> (-13/8,-8/5) Hyperbolic Matrix(17,54,-6,-19) (-7/2,-3/1) -> (-3/1,-11/4) Parabolic Matrix(139,378,82,223) (-11/4,-19/7) -> (5/3,17/10) Hyperbolic Matrix(359,972,106,287) (-19/7,-27/10) -> (27/8,17/5) Hyperbolic Matrix(181,486,54,145) (-27/10,-8/3) -> (10/3,27/8) Hyperbolic Matrix(41,108,-30,-79) (-8/3,-5/2) -> (-11/8,-4/3) Hyperbolic Matrix(155,378,98,239) (-5/2,-17/7) -> (11/7,19/12) Hyperbolic Matrix(23,54,20,47) (-12/5,-7/3) -> (1/1,6/5) Hyperbolic Matrix(47,108,10,23) (-7/3,-9/4) -> (9/2,5/1) Hyperbolic Matrix(25,54,6,13) (-9/4,-2/1) -> (4/1,9/2) Hyperbolic Matrix(29,54,22,41) (-2/1,-9/5) -> (9/7,4/3) Hyperbolic Matrix(61,108,48,85) (-9/5,-7/4) -> (5/4,9/7) Hyperbolic Matrix(31,54,4,7) (-7/4,-12/7) -> (6/1,1/0) Hyperbolic Matrix(95,162,-78,-133) (-12/7,-5/3) -> (-11/9,-6/5) Hyperbolic Matrix(197,324,76,125) (-5/3,-18/11) -> (18/7,13/5) Hyperbolic Matrix(199,324,78,127) (-18/11,-13/8) -> (5/2,18/7) Hyperbolic Matrix(35,54,-24,-37) (-8/5,-3/2) -> (-3/2,-10/7) Parabolic Matrix(341,486,214,305) (-10/7,-27/19) -> (27/17,8/5) Hyperbolic Matrix(685,972,432,613) (-27/19,-17/12) -> (19/12,27/17) Hyperbolic Matrix(115,162,22,31) (-17/12,-7/5) -> (5/1,11/2) Hyperbolic Matrix(233,324,64,89) (-7/5,-18/13) -> (18/5,11/3) Hyperbolic Matrix(235,324,66,91) (-18/13,-11/8) -> (7/2,18/5) Hyperbolic Matrix(41,54,22,29) (-4/3,-9/7) -> (9/5,2/1) Hyperbolic Matrix(85,108,48,61) (-9/7,-5/4) -> (7/4,9/5) Hyperbolic Matrix(131,162,38,47) (-5/4,-11/9) -> (17/5,7/2) Hyperbolic Matrix(47,54,20,23) (-6/5,-1/1) -> (7/3,12/5) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(133,-162,78,-95) (6/5,5/4) -> (17/10,12/7) Hyperbolic Matrix(79,-108,30,-41) (4/3,7/5) -> (13/5,8/3) Hyperbolic Matrix(37,-54,24,-35) (7/5,3/2) -> (3/2,11/7) Parabolic Matrix(67,-108,18,-29) (8/5,5/3) -> (11/3,4/1) Hyperbolic Matrix(67,-162,12,-29) (12/5,5/2) -> (11/2,6/1) Hyperbolic Matrix(19,-54,6,-17) (8/3,3/1) -> (3/1,10/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(7,54,4,31) -> Matrix(1,0,-2,1) Matrix(29,162,-12,-67) -> Matrix(1,0,0,1) Matrix(23,108,10,47) -> Matrix(1,-4,0,1) Matrix(13,54,6,25) -> Matrix(1,2,0,1) Matrix(29,108,-18,-67) -> Matrix(1,2,0,1) Matrix(17,54,-6,-19) -> Matrix(1,0,2,1) Matrix(139,378,82,223) -> Matrix(1,0,-2,1) Matrix(359,972,106,287) -> Matrix(1,0,0,1) Matrix(181,486,54,145) -> Matrix(1,0,0,1) Matrix(41,108,-30,-79) -> Matrix(1,0,-2,1) Matrix(155,378,98,239) -> Matrix(1,-2,0,1) Matrix(23,54,20,47) -> Matrix(1,-2,0,1) Matrix(47,108,10,23) -> Matrix(1,-4,0,1) Matrix(25,54,6,13) -> Matrix(1,2,0,1) Matrix(29,54,22,41) -> Matrix(1,0,2,1) Matrix(61,108,48,85) -> Matrix(1,0,-4,1) Matrix(31,54,4,7) -> Matrix(1,0,-2,1) Matrix(95,162,-78,-133) -> Matrix(1,0,0,1) Matrix(197,324,76,125) -> Matrix(3,-2,-4,3) Matrix(199,324,78,127) -> Matrix(3,-4,-2,3) Matrix(35,54,-24,-37) -> Matrix(1,-2,0,1) Matrix(341,486,214,305) -> Matrix(1,0,0,1) Matrix(685,972,432,613) -> Matrix(1,0,0,1) Matrix(115,162,22,31) -> Matrix(1,0,0,1) Matrix(233,324,64,89) -> Matrix(3,4,2,3) Matrix(235,324,66,91) -> Matrix(3,2,4,3) Matrix(41,54,22,29) -> Matrix(1,0,2,1) Matrix(85,108,48,61) -> Matrix(1,0,-4,1) Matrix(131,162,38,47) -> Matrix(1,0,0,1) Matrix(47,54,20,23) -> Matrix(1,-2,0,1) Matrix(1,0,2,1) -> Matrix(1,0,0,1) Matrix(133,-162,78,-95) -> Matrix(1,0,0,1) Matrix(79,-108,30,-41) -> Matrix(1,0,-2,1) Matrix(37,-54,24,-35) -> Matrix(1,-2,0,1) Matrix(67,-108,18,-29) -> Matrix(1,2,0,1) Matrix(67,-162,12,-29) -> Matrix(1,0,0,1) Matrix(19,-54,6,-17) -> 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): 9 Degree of the the map X: 9 Degree of the the map Y: 36 Permutation triple for Y: ((2,6,23,16,15,11,31,36,26,19,5,18,13,4,3,12,27,8,21,20,35,34,14,10,9,24,7)(17,29,22)(25,33,28); (1,4,16,25,24,9,29,20,19,30,14,13,33,36,31,22,6,21,32,15,34,28,27,12,17,5,2)(3,10,11)(7,26,8); (1,2,8,28,13,18,17,31,10,30,19,7,25,34,35,29,12,11,32,21,26,33,16,23,22,9,3)(4,14,15)(5,20,6)) ----------------------------------------------------------------------- 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. ----------------------------------------------------------------------- INFORMATION ON THE IMAGE IN PSL(2,Z) OF THE GROUP OF MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE Index in PSL(2,Z): 36 Minimal number of generators: 7 Number of equivalence classes of elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 6 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 3/2 9/4 3/1 9/2 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 -1/1 1/1 1/1 0/1 4/3 1/1 3/2 1/0 5/3 0/1 2/1 1/1 9/4 1/0 7/3 -2/1 5/2 1/0 3/1 0/1 4/1 1/1 9/2 1/0 5/1 -2/1 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,1,1) (0/1,1/0) -> (0/1,1/1) Parabolic Matrix(22,-27,9,-11) (1/1,4/3) -> (7/3,5/2) Hyperbolic Matrix(19,-27,12,-17) (4/3,3/2) -> (3/2,5/3) Parabolic Matrix(16,-27,3,-5) (5/3,2/1) -> (5/1,1/0) Hyperbolic Matrix(37,-81,16,-35) (2/1,9/4) -> (9/4,7/3) Parabolic Matrix(10,-27,3,-8) (5/2,3/1) -> (3/1,4/1) Parabolic Matrix(19,-81,4,-17) (4/1,9/2) -> (9/2,5/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(0,-1,1,0) Matrix(22,-27,9,-11) -> Matrix(1,1,-1,0) Matrix(19,-27,12,-17) -> Matrix(1,-1,0,1) Matrix(16,-27,3,-5) -> Matrix(1,1,-1,0) Matrix(37,-81,16,-35) -> Matrix(1,-3,0,1) Matrix(10,-27,3,-8) -> Matrix(1,0,1,1) Matrix(19,-81,4,-17) -> Matrix(1,-3,0,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 1 Minimal number of generators: 2 Number of equivalence classes of elliptic points of order 2: 1 Number of equivalence classes of elliptic points of order 3: 1 Number of equivalence classes of cusps: 1 Genus: 0 Degree of H/liftables -> H/(image of liftables): 9 ----------------------------------------------------------------------- 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 (-1/1,1/1).(0/1,1/0) 0 1 2/1 1/1 1 27 5/2 1/0 1 27 3/1 0/1 3 9 4/1 1/1 1 27 9/2 1/0 9 3 5/1 -2/1 1 27 1/0 1/0 1 27 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(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) Reflection Matrix(11,-27,2,-5) (2/1,5/2) -> (5/1,1/0) Glide Reflection Matrix(10,-27,3,-8) (5/2,3/1) -> (3/1,4/1) Parabolic Matrix(19,-81,4,-17) (4/1,9/2) -> (9/2,5/1) Parabolic IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,1,-1) -> Matrix(0,1,1,0) (0/1,2/1) -> (-1/1,1/1) Matrix(11,-27,2,-5) -> Matrix(1,1,0,-1) *** -> (-1/2,1/0) Matrix(10,-27,3,-8) -> Matrix(1,0,1,1) 0/1 Matrix(19,-81,4,-17) -> Matrix(1,-3,0,1) 1/0 ----------------------------------------------------------------------- The pullback map has no extra symmetries.