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 -5/1 -4/1 -3/1 -5/2 -2/1 -1/1 -1/2 -2/5 -1/4 0/1 1/5 1/4 1/2 4/5 1/1 5/4 7/5 3/2 2/1 3/1 4/1 5/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 1/2 1/0 -4/1 1/1 -11/3 3/2 1/0 -7/2 1/1 -3/1 1/0 -8/3 0/1 2/1 -5/2 1/0 -7/3 -1/2 1/0 -9/4 -1/1 1/1 -2/1 0/1 -1/1 1/2 1/0 -2/3 0/1 -9/14 1/1 -7/11 1/2 1/0 -5/8 1/1 -3/5 1/0 -4/7 0/1 -5/9 1/2 1/0 -1/2 1/1 -4/9 0/1 2/1 -3/7 3/2 1/0 -2/5 1/0 -3/8 -1/1 1/1 -4/11 0/1 2/1 -1/3 1/0 -3/10 -1/1 -2/7 0/1 -1/4 0/1 -2/9 0/1 -3/14 1/1 -1/5 1/2 1/0 0/1 0/1 2/1 1/5 1/0 2/9 0/1 1/4 -1/1 1/1 2/7 1/0 1/3 -1/2 1/0 1/2 0/1 3/5 1/4 1/2 5/8 1/3 1/1 2/3 0/1 5/7 1/4 1/2 3/4 1/3 1/1 7/9 1/2 1/0 4/5 0/1 1/1 1/2 5/4 1/1 9/7 1/2 1/0 4/3 0/1 2/3 11/8 3/5 1/1 7/5 1/2 3/4 10/7 2/3 3/2 1/1 2/1 1/1 5/2 1/1 18/7 4/3 13/5 3/2 8/3 4/3 2/1 3/1 3/2 1/0 7/2 1/0 4/1 0/1 2/1 9/2 1/1 5/1 1/0 11/2 -1/1 6/1 0/1 1/0 -1/1 1/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,28,8,45) (-5/1,1/0) -> (3/5,5/8) Hyperbolic Matrix(15,64,-4,-17) (-5/1,-4/1) -> (-4/1,-11/3) Parabolic Matrix(41,148,-64,-231) (-11/3,-7/2) -> (-9/14,-7/11) Hyperbolic Matrix(5,16,-16,-51) (-7/2,-3/1) -> (-1/3,-3/10) Hyperbolic Matrix(7,20,-20,-57) (-3/1,-8/3) -> (-4/11,-1/3) Hyperbolic Matrix(29,76,8,21) (-8/3,-5/2) -> (7/2,4/1) Hyperbolic Matrix(27,64,8,19) (-5/2,-7/3) -> (3/1,7/2) Hyperbolic Matrix(37,84,48,109) (-7/3,-9/4) -> (3/4,7/9) Hyperbolic Matrix(9,20,40,89) (-9/4,-2/1) -> (2/9,1/4) Hyperbolic Matrix(3,4,-4,-5) (-2/1,-1/1) -> (-1/1,-2/3) Parabolic Matrix(267,172,104,67) (-2/3,-9/14) -> (5/2,18/7) Hyperbolic Matrix(127,80,100,63) (-7/11,-5/8) -> (5/4,9/7) Hyperbolic Matrix(33,20,28,17) (-5/8,-3/5) -> (1/1,5/4) Hyperbolic Matrix(27,16,32,19) (-3/5,-4/7) -> (4/5,1/1) Hyperbolic Matrix(85,48,108,61) (-4/7,-5/9) -> (7/9,4/5) Hyperbolic Matrix(29,16,-136,-75) (-5/9,-1/2) -> (-3/14,-1/5) Hyperbolic Matrix(89,40,20,9) (-1/2,-4/9) -> (4/1,9/2) Hyperbolic Matrix(109,48,84,37) (-4/9,-3/7) -> (9/7,4/3) Hyperbolic Matrix(19,8,64,27) (-3/7,-2/5) -> (2/7,1/3) Hyperbolic Matrix(21,8,76,29) (-2/5,-3/8) -> (1/4,2/7) Hyperbolic Matrix(153,56,112,41) (-3/8,-4/11) -> (4/3,11/8) Hyperbolic Matrix(121,36,84,25) (-3/10,-2/7) -> (10/7,3/2) Hyperbolic Matrix(15,4,-64,-17) (-2/7,-1/4) -> (-1/4,-2/9) Parabolic Matrix(183,40,32,7) (-2/9,-3/14) -> (11/2,6/1) Hyperbolic Matrix(43,8,16,3) (-1/5,0/1) -> (8/3,3/1) Hyperbolic Matrix(53,-8,20,-3) (0/1,1/5) -> (13/5,8/3) Hyperbolic Matrix(207,-44,80,-17) (1/5,2/9) -> (18/7,13/5) Hyperbolic Matrix(9,-4,16,-7) (1/3,1/2) -> (1/2,3/5) Parabolic Matrix(51,-32,8,-5) (5/8,2/3) -> (6/1,1/0) Hyperbolic Matrix(91,-64,64,-45) (2/3,5/7) -> (7/5,10/7) Hyperbolic Matrix(105,-76,76,-55) (5/7,3/4) -> (11/8,7/5) Hyperbolic Matrix(9,-16,4,-7) (3/2,2/1) -> (2/1,5/2) Parabolic Matrix(21,-100,4,-19) (9/2,5/1) -> (5/1,11/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(5,28,8,45) -> Matrix(1,0,2,1) Matrix(15,64,-4,-17) -> Matrix(3,-2,2,-1) Matrix(41,148,-64,-231) -> Matrix(1,-2,2,-3) Matrix(5,16,-16,-51) -> Matrix(1,-2,0,1) Matrix(7,20,-20,-57) -> Matrix(1,0,0,1) Matrix(29,76,8,21) -> Matrix(1,0,0,1) Matrix(27,64,8,19) -> Matrix(1,2,0,1) Matrix(37,84,48,109) -> Matrix(1,0,2,1) Matrix(9,20,40,89) -> Matrix(1,0,0,1) Matrix(3,4,-4,-5) -> Matrix(1,0,0,1) Matrix(267,172,104,67) -> Matrix(5,-4,4,-3) Matrix(127,80,100,63) -> Matrix(1,0,0,1) Matrix(33,20,28,17) -> Matrix(1,-2,2,-3) Matrix(27,16,32,19) -> Matrix(1,0,2,1) Matrix(85,48,108,61) -> Matrix(1,0,0,1) Matrix(29,16,-136,-75) -> Matrix(1,0,0,1) Matrix(89,40,20,9) -> Matrix(1,0,0,1) Matrix(109,48,84,37) -> Matrix(1,-2,2,-3) Matrix(19,8,64,27) -> Matrix(1,-2,0,1) Matrix(21,8,76,29) -> Matrix(1,0,0,1) Matrix(153,56,112,41) -> Matrix(1,-2,2,-3) Matrix(121,36,84,25) -> Matrix(3,2,4,3) Matrix(15,4,-64,-17) -> Matrix(1,0,2,1) Matrix(183,40,32,7) -> Matrix(1,0,-2,1) Matrix(43,8,16,3) -> Matrix(3,-2,2,-1) Matrix(53,-8,20,-3) -> Matrix(3,-2,2,-1) Matrix(207,-44,80,-17) -> Matrix(3,4,2,3) Matrix(9,-4,16,-7) -> Matrix(1,0,4,1) Matrix(51,-32,8,-5) -> Matrix(1,0,-2,1) Matrix(91,-64,64,-45) -> Matrix(5,-2,8,-3) Matrix(105,-76,76,-55) -> Matrix(5,-2,8,-3) Matrix(9,-16,4,-7) -> Matrix(5,-4,4,-3) Matrix(21,-100,4,-19) -> Matrix(1,-2,0,1) 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): 6 Degree of the the map X: 6 Degree of the the map Y: 32 Permutation triple for Y: ((1,7,22,15,8,2)(3,10,28,32,29,11)(4,5)(6,19,24,23,31,20)(9,21)(12,30)(13,26,17,16,25,14)(18,27); (1,5,17,10,18,6)(2,3)(4,14,29,27,23,15)(7,20,30,28,26,21)(8,24,12,11,25,9)(13,31)(16,19)(22,32); (1,3,12,20,13,4)(2,9,26,31,27,10)(5,15,32,30,24,16)(6,7)(8,23)(11,14)(17,28)(18,29,22,21,25,19)) ----------------------------------------------------------------------- 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 -5/1 0 6 -4/1 1/1 1 2 -11/3 0 6 -7/2 1/1 1 6 -3/1 1/0 1 6 -8/3 0 6 -5/2 1/0 1 2 -7/3 0 6 -9/4 0 6 -2/1 0/1 1 6 -11/6 1/1 1 6 -9/5 0 6 -7/4 1/1 1 2 -5/3 1/0 1 6 -1/1 (1/2,1/0) 0 2 1/1 1/2 1 6 5/4 1/1 1 2 9/7 0 6 4/3 0 6 7/5 (1/2,3/4) 0 2 3/2 1/1 1 6 2/1 1/1 2 2 5/2 1/1 1 6 8/3 0 6 3/1 0 6 7/2 1/0 1 2 4/1 0 6 9/2 1/1 1 6 5/1 1/0 1 2 11/2 -1/1 1 6 6/1 0/1 1 6 1/0 0 6 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(8,43,3,16) (-5/1,1/0) -> (8/3,3/1) Glide Reflection Matrix(15,64,-4,-17) (-5/1,-4/1) -> (-4/1,-11/3) Parabolic Matrix(51,184,-28,-101) (-11/3,-7/2) -> (-11/6,-9/5) Hyperbolic Matrix(29,100,20,69) (-7/2,-10/3) -> (10/7,3/2) Hyperbolic Matrix(16,51,-5,-16) (-17/5,-3/1) -> (-17/5,-3/1) Reflection Matrix(20,57,-7,-20) (-3/1,-19/7) -> (-3/1,-19/7) Reflection Matrix(49,132,36,97) (-11/4,-8/3) -> (4/3,11/8) Hyperbolic Matrix(29,76,8,21) (-8/3,-5/2) -> (7/2,4/1) Hyperbolic Matrix(27,64,8,19) (-5/2,-7/3) -> (3/1,7/2) Hyperbolic Matrix(48,109,37,84) (-7/3,-9/4) -> (9/7,4/3) Glide Reflection Matrix(40,89,9,20) (-9/4,-2/1) -> (4/1,9/2) Glide Reflection Matrix(72,133,13,24) (-2/1,-11/6) -> (11/2,6/1) Glide Reflection Matrix(61,108,48,85) (-9/5,-7/4) -> (5/4,9/7) Hyperbolic Matrix(19,32,16,27) (-7/4,-5/3) -> (1/1,5/4) Hyperbolic Matrix(4,5,-3,-4) (-5/3,-1/1) -> (-5/3,-1/1) Reflection Matrix(0,1,1,0) (-1/1,1/1) -> (-1/1,1/1) Reflection Matrix(76,-105,55,-76) (15/11,7/5) -> (15/11,7/5) Reflection Matrix(64,-91,45,-64) (7/5,13/9) -> (7/5,13/9) Reflection Matrix(9,-16,4,-7) (3/2,2/1) -> (2/1,5/2) Parabolic Matrix(20,-53,3,-8) (5/2,8/3) -> (6/1,1/0) Glide Reflection Matrix(21,-100,4,-19) (9/2,5/1) -> (5/1,11/2) Parabolic IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(8,43,3,16) -> Matrix(3,-1,2,-1) Matrix(15,64,-4,-17) -> Matrix(3,-2,2,-1) 1/1 Matrix(51,184,-28,-101) -> Matrix(1,-2,2,-3) 1/1 Matrix(29,100,20,69) -> Matrix(3,-4,4,-5) 1/1 Matrix(16,51,-5,-16) -> Matrix(-1,3,0,1) (-17/5,-3/1) -> (3/2,1/0) Matrix(20,57,-7,-20) -> Matrix(-1,1,0,1) (-3/1,-19/7) -> (1/2,1/0) Matrix(49,132,36,97) -> Matrix(1,-2,2,-3) 1/1 Matrix(29,76,8,21) -> Matrix(1,0,0,1) Matrix(27,64,8,19) -> Matrix(1,2,0,1) 1/0 Matrix(48,109,37,84) -> Matrix(1,1,2,1) Matrix(40,89,9,20) -> Matrix(-1,1,0,1) *** -> (1/2,1/0) Matrix(72,133,13,24) -> Matrix(1,-1,-2,1) Matrix(61,108,48,85) -> Matrix(1,0,0,1) Matrix(19,32,16,27) -> Matrix(1,-2,2,-3) 1/1 Matrix(4,5,-3,-4) -> Matrix(-1,1,0,1) (-5/3,-1/1) -> (1/2,1/0) Matrix(0,1,1,0) -> Matrix(-1,1,0,1) (-1/1,1/1) -> (1/2,1/0) Matrix(76,-105,55,-76) -> Matrix(5,-3,8,-5) (15/11,7/5) -> (1/2,3/4) Matrix(64,-91,45,-64) -> Matrix(5,-3,8,-5) (7/5,13/9) -> (1/2,3/4) Matrix(9,-16,4,-7) -> Matrix(5,-4,4,-3) 1/1 Matrix(20,-53,3,-8) -> Matrix(1,-1,2,-3) Matrix(21,-100,4,-19) -> Matrix(1,-2,0,1) 1/0 ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.