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 -3/2 -1/1 -3/4 -1/2 -7/16 -3/7 -1/3 -1/4 -1/5 0/1 1/5 1/4 1/3 3/8 3/7 1/2 5/8 2/3 3/4 1/1 4/3 3/2 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 -1/1 -5/3 -1/2 -8/5 1/1 -11/7 -2/1 1/0 -3/2 -1/1 -13/9 -1/2 0/1 -10/7 -1/1 -7/5 -1/2 -4/3 -1/1 -1/1 -1/1 0/1 -4/5 -1/1 -3/4 -1/1 -8/11 -1/1 -5/7 -1/2 -2/3 -1/1 -5/8 0/1 -8/13 1/1 -3/5 1/0 -4/7 -1/1 -1/2 -1/1 -4/9 -1/1 -7/16 -2/3 -10/23 -3/5 -3/7 -2/3 -1/2 -2/5 -1/3 -3/8 0/1 -4/11 1/3 -1/3 1/0 -2/7 -1/1 -1/4 -1/1 -2/9 -1/1 -1/5 -1/1 -2/3 0/1 -1/1 1/5 -1/2 1/4 -1/1 1/3 -1/1 -2/3 3/8 -2/3 2/5 -3/5 3/7 -1/2 1/2 -1/2 5/9 -1/2 9/16 0/1 4/7 -1/1 3/5 -1/1 -2/3 5/8 -2/3 2/3 -3/5 5/7 -6/11 -1/2 3/4 -1/2 1/1 -1/2 5/4 -1/2 9/7 -1/2 -6/13 4/3 -3/7 11/8 -2/5 7/5 -2/5 -1/3 10/7 -1/3 3/2 -1/2 14/9 -3/7 25/16 -2/5 11/7 -1/2 8/5 -3/7 13/8 -2/5 5/3 -2/5 -1/3 7/4 -1/3 9/5 -1/2 2/1 -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(7,12,-24,-41) (-2/1,-5/3) -> (-1/3,-2/7) Hyperbolic Matrix(17,28,-48,-79) (-5/3,-8/5) -> (-4/11,-1/3) Hyperbolic Matrix(73,116,56,89) (-8/5,-11/7) -> (9/7,4/3) Hyperbolic Matrix(47,72,-32,-49) (-11/7,-3/2) -> (-3/2,-13/9) Parabolic Matrix(111,160,-256,-369) (-13/9,-10/7) -> (-10/23,-3/7) Hyperbolic Matrix(73,104,40,57) (-10/7,-7/5) -> (9/5,2/1) Hyperbolic Matrix(49,68,-80,-111) (-7/5,-4/3) -> (-8/13,-3/5) Hyperbolic Matrix(7,8,-8,-9) (-4/3,-1/1) -> (-1/1,-4/5) Parabolic Matrix(47,36,-64,-49) (-4/5,-3/4) -> (-3/4,-8/11) Parabolic Matrix(177,128,112,81) (-8/11,-5/7) -> (11/7,8/5) Hyperbolic Matrix(23,16,56,39) (-5/7,-2/3) -> (2/5,3/7) Hyperbolic Matrix(31,20,48,31) (-2/3,-5/8) -> (5/8,2/3) Hyperbolic Matrix(175,108,128,79) (-5/8,-8/13) -> (4/3,11/8) Hyperbolic Matrix(7,4,40,23) (-3/5,-4/7) -> (0/1,1/5) Hyperbolic Matrix(15,8,-32,-17) (-4/7,-1/2) -> (-1/2,-4/9) Parabolic Matrix(145,64,256,113) (-4/9,-7/16) -> (9/16,4/7) Hyperbolic Matrix(799,348,512,223) (-7/16,-10/23) -> (14/9,25/16) Hyperbolic Matrix(39,16,56,23) (-3/7,-2/5) -> (2/3,5/7) Hyperbolic Matrix(31,12,80,31) (-2/5,-3/8) -> (3/8,2/5) Hyperbolic Matrix(207,76,128,47) (-3/8,-4/11) -> (8/5,13/8) Hyperbolic Matrix(15,4,-64,-17) (-2/7,-1/4) -> (-1/4,-2/9) Parabolic Matrix(113,24,80,17) (-2/9,-1/5) -> (7/5,10/7) Hyperbolic Matrix(23,4,40,7) (-1/5,0/1) -> (4/7,3/5) Hyperbolic Matrix(71,-16,40,-9) (1/5,1/4) -> (7/4,9/5) Hyperbolic Matrix(41,-12,24,-7) (1/4,1/3) -> (5/3,7/4) Hyperbolic Matrix(79,-28,48,-17) (1/3,3/8) -> (13/8,5/3) Hyperbolic Matrix(17,-8,32,-15) (3/7,1/2) -> (1/2,5/9) Parabolic Matrix(401,-224,256,-143) (5/9,9/16) -> (25/16,11/7) Hyperbolic Matrix(111,-68,80,-49) (3/5,5/8) -> (11/8,7/5) Hyperbolic Matrix(71,-52,56,-41) (5/7,3/4) -> (5/4,9/7) Hyperbolic Matrix(9,-8,8,-7) (3/4,1/1) -> (1/1,5/4) Parabolic Matrix(49,-72,32,-47) (10/7,3/2) -> (3/2,14/9) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,0,-2,1) Matrix(7,12,-24,-41) -> Matrix(3,2,-2,-1) Matrix(17,28,-48,-79) -> Matrix(1,0,2,1) Matrix(73,116,56,89) -> Matrix(1,-4,-2,9) Matrix(47,72,-32,-49) -> Matrix(1,2,-2,-3) Matrix(111,160,-256,-369) -> Matrix(5,2,-8,-3) Matrix(73,104,40,57) -> Matrix(3,2,-8,-5) Matrix(49,68,-80,-111) -> Matrix(1,0,2,1) Matrix(7,8,-8,-9) -> Matrix(1,0,0,1) Matrix(47,36,-64,-49) -> Matrix(1,2,-2,-3) Matrix(177,128,112,81) -> Matrix(7,4,-16,-9) Matrix(23,16,56,39) -> Matrix(5,2,-8,-3) Matrix(31,20,48,31) -> Matrix(5,2,-8,-3) Matrix(175,108,128,79) -> Matrix(5,-2,-12,5) Matrix(7,4,40,23) -> Matrix(1,2,-2,-3) Matrix(15,8,-32,-17) -> Matrix(1,2,-2,-3) Matrix(145,64,256,113) -> Matrix(3,2,-2,-1) Matrix(799,348,512,223) -> Matrix(19,12,-46,-29) Matrix(39,16,56,23) -> Matrix(9,4,-16,-7) Matrix(31,12,80,31) -> Matrix(9,2,-14,-3) Matrix(207,76,128,47) -> Matrix(9,-2,-22,5) Matrix(15,4,-64,-17) -> Matrix(5,6,-6,-7) Matrix(113,24,80,17) -> Matrix(5,4,-14,-11) Matrix(23,4,40,7) -> Matrix(1,0,0,1) Matrix(71,-16,40,-9) -> Matrix(3,2,-8,-5) Matrix(41,-12,24,-7) -> Matrix(5,4,-14,-11) Matrix(79,-28,48,-17) -> Matrix(5,4,-14,-11) Matrix(17,-8,32,-15) -> Matrix(3,2,-8,-5) Matrix(401,-224,256,-143) -> Matrix(3,2,-8,-5) Matrix(111,-68,80,-49) -> Matrix(5,4,-14,-11) Matrix(71,-52,56,-41) -> Matrix(23,12,-48,-25) Matrix(9,-8,8,-7) -> Matrix(3,2,-8,-5) Matrix(49,-72,32,-47) -> 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): 11 Degree of the the map X: 11 Degree of the the map Y: 32 Permutation triple for Y: ((1,6,22,25,31,19,7,2)(3,12,23,30,32,28,13,4)(5,17,8,21,20,10,9,18)(11,27,14,24,29,16,15,26); (1,4,16,5)(2,10,11,3)(6,21,14,13)(7,12,24,8)(9,22,28,26)(15,23,19,18)(17,29,32,25)(20,31,30,27); (1,3)(2,8,25,9)(4,14,30,15)(5,19,20,6)(7,23)(10,27)(11,28,29,12)(13,22)(16,17)(18,26)(21,24)(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 (-1/1,0/1) 0 8 -3/4 -1/1 1 2 -5/7 -1/2 2 8 -2/3 -1/1 1 8 -5/8 0/1 3 2 -3/5 1/0 2 8 -4/7 -1/1 1 8 -1/2 -1/1 1 4 -4/9 -1/1 1 8 -7/16 -2/3 2 2 -3/7 (-2/3,-1/2) 0 8 -2/5 -1/3 1 8 -3/8 0/1 5 2 -1/3 1/0 2 8 -1/4 -1/1 3 2 -1/5 (-1/1,-2/3) 0 8 0/1 -1/1 1 8 1/5 -1/2 2 8 1/4 -1/1 1 2 1/3 (-1/1,-2/3) 0 8 3/8 -2/3 5 2 2/5 -3/5 1 8 3/7 -1/2 2 8 1/2 -1/2 1 4 5/9 -1/2 2 8 9/16 0/1 2 2 4/7 -1/1 1 8 3/5 (-1/1,-2/3) 0 8 5/8 -2/3 3 2 2/3 -3/5 1 8 5/7 (-6/11,-1/2) 0 8 3/4 -1/2 5 2 1/1 -1/2 2 8 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(7,6,-8,-7) (-1/1,-3/4) -> (-1/1,-3/4) Reflection Matrix(41,30,-56,-41) (-3/4,-5/7) -> (-3/4,-5/7) Reflection Matrix(23,16,56,39) (-5/7,-2/3) -> (2/5,3/7) Hyperbolic Matrix(31,20,48,31) (-2/3,-5/8) -> (5/8,2/3) Hyperbolic Matrix(49,30,-80,-49) (-5/8,-3/5) -> (-5/8,-3/5) Reflection Matrix(7,4,40,23) (-3/5,-4/7) -> (0/1,1/5) Hyperbolic Matrix(15,8,-32,-17) (-4/7,-1/2) -> (-1/2,-4/9) Parabolic Matrix(145,64,256,113) (-4/9,-7/16) -> (9/16,4/7) Hyperbolic Matrix(97,42,-224,-97) (-7/16,-3/7) -> (-7/16,-3/7) Reflection Matrix(39,16,56,23) (-3/7,-2/5) -> (2/3,5/7) Hyperbolic Matrix(31,12,80,31) (-2/5,-3/8) -> (3/8,2/5) Hyperbolic Matrix(17,6,-48,-17) (-3/8,-1/3) -> (-3/8,-1/3) Reflection Matrix(7,2,-24,-7) (-1/3,-1/4) -> (-1/3,-1/4) Reflection Matrix(9,2,-40,-9) (-1/4,-1/5) -> (-1/4,-1/5) Reflection Matrix(23,4,40,7) (-1/5,0/1) -> (4/7,3/5) Hyperbolic Matrix(9,-2,40,-9) (1/5,1/4) -> (1/5,1/4) Reflection Matrix(7,-2,24,-7) (1/4,1/3) -> (1/4,1/3) Reflection Matrix(17,-6,48,-17) (1/3,3/8) -> (1/3,3/8) Reflection Matrix(17,-8,32,-15) (3/7,1/2) -> (1/2,5/9) Parabolic Matrix(161,-90,288,-161) (5/9,9/16) -> (5/9,9/16) Reflection Matrix(49,-30,80,-49) (3/5,5/8) -> (3/5,5/8) Reflection Matrix(41,-30,56,-41) (5/7,3/4) -> (5/7,3/4) Reflection Matrix(7,-6,8,-7) (3/4,1/1) -> (3/4,1/1) 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) -> (-1/1,0/1) Matrix(7,6,-8,-7) -> Matrix(-1,0,2,1) (-1/1,-3/4) -> (-1/1,0/1) Matrix(41,30,-56,-41) -> Matrix(3,2,-4,-3) (-3/4,-5/7) -> (-1/1,-1/2) Matrix(23,16,56,39) -> Matrix(5,2,-8,-3) -1/2 Matrix(31,20,48,31) -> Matrix(5,2,-8,-3) -1/2 Matrix(49,30,-80,-49) -> Matrix(1,0,0,-1) (-5/8,-3/5) -> (0/1,1/0) Matrix(7,4,40,23) -> Matrix(1,2,-2,-3) -1/1 Matrix(15,8,-32,-17) -> Matrix(1,2,-2,-3) -1/1 Matrix(145,64,256,113) -> Matrix(3,2,-2,-1) -1/1 Matrix(97,42,-224,-97) -> Matrix(7,4,-12,-7) (-7/16,-3/7) -> (-2/3,-1/2) Matrix(39,16,56,23) -> Matrix(9,4,-16,-7) -1/2 Matrix(31,12,80,31) -> Matrix(9,2,-14,-3) Matrix(17,6,-48,-17) -> Matrix(1,0,0,-1) (-3/8,-1/3) -> (0/1,1/0) Matrix(7,2,-24,-7) -> Matrix(1,2,0,-1) (-1/3,-1/4) -> (-1/1,1/0) Matrix(9,2,-40,-9) -> Matrix(5,4,-6,-5) (-1/4,-1/5) -> (-1/1,-2/3) Matrix(23,4,40,7) -> Matrix(1,0,0,1) Matrix(9,-2,40,-9) -> Matrix(3,2,-4,-3) (1/5,1/4) -> (-1/1,-1/2) Matrix(7,-2,24,-7) -> Matrix(5,4,-6,-5) (1/4,1/3) -> (-1/1,-2/3) Matrix(17,-6,48,-17) -> Matrix(5,4,-6,-5) (1/3,3/8) -> (-1/1,-2/3) Matrix(17,-8,32,-15) -> Matrix(3,2,-8,-5) -1/2 Matrix(161,-90,288,-161) -> Matrix(-1,0,4,1) (5/9,9/16) -> (-1/2,0/1) Matrix(49,-30,80,-49) -> Matrix(5,4,-6,-5) (3/5,5/8) -> (-1/1,-2/3) Matrix(41,-30,56,-41) -> Matrix(23,12,-44,-23) (5/7,3/4) -> (-6/11,-1/2) Matrix(7,-6,8,-7) -> Matrix(3,2,-4,-3) (3/4,1/1) -> (-1/1,-1/2) Matrix(-1,2,0,1) -> Matrix(-1,0,4,1) (1/1,1/0) -> (-1/2,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.