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 -3/1 -2/1 -1/1 -1/2 -3/7 -1/3 -1/4 -1/5 0/1 1/4 1/3 1/2 3/5 3/4 1/1 9/7 3/2 5/3 2/1 5/2 3/1 11/3 4/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 1/0 -7/2 0/1 1/0 -10/3 1/2 -3/1 0/1 -14/5 1/4 -11/4 1/3 1/2 -8/3 1/2 -5/2 1/2 1/1 -2/1 1/0 -1/1 0/1 -2/3 1/2 -5/8 0/1 1/4 -8/13 1/4 -3/5 1/3 -7/12 1/3 2/5 -4/7 1/2 -1/2 0/1 1/2 -4/9 1/2 -3/7 1/2 -2/5 1/2 -3/8 2/3 1/1 -1/3 1/1 -2/7 1/0 -1/4 0/1 1/0 -2/9 -1/2 -1/5 0/1 0/1 1/2 1/5 1/1 1/4 2/3 1/1 1/3 1/1 3/8 1/1 4/3 5/13 1/1 2/5 3/2 1/2 1/1 1/0 4/7 1/0 3/5 1/0 8/13 1/0 5/8 0/1 1/0 2/3 1/0 5/7 1/0 8/11 1/0 3/4 -1/1 1/0 1/1 0/1 5/4 1/2 1/1 14/11 3/4 9/7 1/1 4/3 1/0 7/5 0/1 10/7 1/4 3/2 0/1 1/2 8/5 1/2 5/3 0/1 12/7 1/4 7/4 0/1 1/3 2/1 1/2 7/3 1/1 12/5 1/2 17/7 1/1 5/2 1/2 1/1 3/1 1/1 7/2 1/1 1/0 25/7 1/1 18/5 1/0 11/3 1/1 4/1 1/0 1/0 0/1 1/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,8,0,1) (-4/1,1/0) -> (4/1,1/0) Parabolic Matrix(19,68,12,43) (-4/1,-7/2) -> (3/2,8/5) Hyperbolic Matrix(29,100,20,69) (-7/2,-10/3) -> (10/7,3/2) Hyperbolic Matrix(23,72,-8,-25) (-10/3,-3/1) -> (-3/1,-14/5) Parabolic Matrix(81,224,64,177) (-14/5,-11/4) -> (5/4,14/11) Hyperbolic Matrix(41,112,56,153) (-11/4,-8/3) -> (8/11,3/4) Hyperbolic Matrix(11,28,20,51) (-8/3,-5/2) -> (1/2,4/7) Hyperbolic Matrix(5,12,12,29) (-5/2,-2/1) -> (2/5,1/2) Hyperbolic Matrix(3,4,-4,-5) (-2/1,-1/1) -> (-1/1,-2/3) Parabolic Matrix(19,12,-84,-53) (-2/3,-5/8) -> (-1/4,-2/9) Hyperbolic Matrix(129,80,208,129) (-5/8,-8/13) -> (8/13,5/8) Hyperbolic Matrix(163,100,44,27) (-8/13,-3/5) -> (11/3,4/1) Hyperbolic Matrix(75,44,196,115) (-3/5,-7/12) -> (3/8,5/13) Hyperbolic Matrix(193,112,112,65) (-7/12,-4/7) -> (12/7,7/4) Hyperbolic Matrix(15,8,-32,-17) (-4/7,-1/2) -> (-1/2,-4/9) Parabolic Matrix(101,44,140,61) (-4/9,-3/7) -> (5/7,8/11) Hyperbolic Matrix(39,16,56,23) (-3/7,-2/5) -> (2/3,5/7) Hyperbolic Matrix(51,20,28,11) (-2/5,-3/8) -> (7/4,2/1) Hyperbolic Matrix(11,4,52,19) (-3/8,-1/3) -> (1/5,1/4) Hyperbolic Matrix(55,16,24,7) (-1/3,-2/7) -> (2/1,7/3) Hyperbolic Matrix(43,12,68,19) (-2/7,-1/4) -> (5/8,2/3) Hyperbolic Matrix(113,24,80,17) (-2/9,-1/5) -> (7/5,10/7) Hyperbolic Matrix(27,4,20,3) (-1/5,0/1) -> (4/3,7/5) Hyperbolic Matrix(67,-12,28,-5) (0/1,1/5) -> (7/3,12/5) Hyperbolic Matrix(13,-4,36,-11) (1/4,1/3) -> (1/3,3/8) Parabolic Matrix(289,-112,80,-31) (5/13,2/5) -> (18/5,11/3) Hyperbolic Matrix(61,-36,100,-59) (4/7,3/5) -> (3/5,8/13) Parabolic Matrix(9,-8,8,-7) (3/4,1/1) -> (1/1,5/4) Parabolic Matrix(401,-512,112,-143) (14/11,9/7) -> (25/7,18/5) Hyperbolic Matrix(135,-176,56,-73) (9/7,4/3) -> (12/5,17/7) Hyperbolic Matrix(61,-100,36,-59) (8/5,5/3) -> (5/3,12/7) Parabolic Matrix(99,-244,28,-69) (17/7,5/2) -> (7/2,25/7) Hyperbolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,8,0,1) -> Matrix(1,0,0,1) Matrix(19,68,12,43) -> Matrix(1,0,2,1) Matrix(29,100,20,69) -> Matrix(1,0,2,1) Matrix(23,72,-8,-25) -> Matrix(1,0,2,1) Matrix(81,224,64,177) -> Matrix(5,-2,8,-3) Matrix(41,112,56,153) -> Matrix(5,-2,-2,1) Matrix(11,28,20,51) -> Matrix(3,-2,2,-1) Matrix(5,12,12,29) -> Matrix(3,-2,2,-1) Matrix(3,4,-4,-5) -> Matrix(1,0,2,1) Matrix(19,12,-84,-53) -> Matrix(1,0,-4,1) Matrix(129,80,208,129) -> Matrix(1,0,-4,1) Matrix(163,100,44,27) -> Matrix(7,-2,4,-1) Matrix(75,44,196,115) -> Matrix(7,-2,4,-1) Matrix(193,112,112,65) -> Matrix(5,-2,18,-7) Matrix(15,8,-32,-17) -> Matrix(1,0,0,1) Matrix(101,44,140,61) -> Matrix(9,-4,-2,1) Matrix(39,16,56,23) -> Matrix(3,-2,2,-1) Matrix(51,20,28,11) -> Matrix(3,-2,8,-5) Matrix(11,4,52,19) -> Matrix(1,0,0,1) Matrix(55,16,24,7) -> Matrix(1,-2,2,-3) Matrix(43,12,68,19) -> Matrix(1,0,0,1) Matrix(113,24,80,17) -> Matrix(1,0,6,1) Matrix(27,4,20,3) -> Matrix(1,0,-2,1) Matrix(67,-12,28,-5) -> Matrix(1,0,0,1) Matrix(13,-4,36,-11) -> Matrix(7,-6,6,-5) Matrix(289,-112,80,-31) -> Matrix(1,-2,2,-3) Matrix(61,-36,100,-59) -> Matrix(1,-2,0,1) Matrix(9,-8,8,-7) -> Matrix(1,0,2,1) Matrix(401,-512,112,-143) -> Matrix(5,-4,4,-3) Matrix(135,-176,56,-73) -> Matrix(1,-2,2,-3) Matrix(61,-100,36,-59) -> Matrix(1,0,2,1) Matrix(99,-244,28,-69) -> Matrix(3,-2,2,-1) Matrix(13,-36,4,-11) -> Matrix(3,-2,2,-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): 8 Degree of the the map X: 8 Degree of the the map Y: 32 Permutation triple for Y: ((1,7,23,31,18,17,8,2)(3,10,24,29,32,30,22,11)(4,14,20,6,19,9,15,5)(12,16,26,25,28,21,27,13); (1,5,18,6)(2,3)(4,13)(7,22)(8,14,23,9)(10,26,30,27)(11,28,29,12)(15,16)(17,24)(19,25)(20,21)(31,32); (1,3,12,4)(2,9,25,10)(5,16,29,17)(6,21,11,7)(8,24,27,20)(13,30,31,14)(15,23,22,26)(18,32,28,19)) ----------------------------------------------------------------------- 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 -1/1 0/1 1 2 -1/2 (0/1,1/2) 0 8 -3/7 1/2 3 2 -2/5 1/2 1 8 -3/8 (2/3,1/1) 0 8 -1/3 1/1 1 4 -2/7 1/0 1 8 -1/4 (0/1,1/0) 0 8 -1/5 0/1 4 2 0/1 1/2 1 8 1/5 1/1 1 4 1/4 (2/3,1/1) 0 8 1/3 1/1 3 2 1/2 (1/1,1/0) 0 8 3/5 1/0 1 2 5/8 (0/1,1/0) 0 8 2/3 1/0 1 8 5/7 1/0 3 2 3/4 (-1/1,1/0) 0 8 1/1 0/1 1 4 5/4 (1/2,1/1) 0 8 9/7 1/1 2 2 4/3 1/0 1 8 7/5 0/1 4 2 3/2 (0/1,1/2) 0 8 5/3 0/1 1 2 7/4 (0/1,1/3) 0 8 2/1 1/2 1 8 7/3 1/1 1 4 12/5 1/2 1 8 17/7 1/1 2 2 5/2 (1/2,1/1) 0 8 3/1 1/1 1 2 1/0 (0/1,1/1) 0 8 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(3,2,-4,-3) (-1/1,-1/2) -> (-1/1,-1/2) Reflection Matrix(13,6,-28,-13) (-1/2,-3/7) -> (-1/2,-3/7) Reflection Matrix(39,16,56,23) (-3/7,-2/5) -> (2/3,5/7) Hyperbolic Matrix(51,20,28,11) (-2/5,-3/8) -> (7/4,2/1) Hyperbolic Matrix(11,4,52,19) (-3/8,-1/3) -> (1/5,1/4) Hyperbolic Matrix(55,16,24,7) (-1/3,-2/7) -> (2/1,7/3) Hyperbolic Matrix(43,12,68,19) (-2/7,-1/4) -> (5/8,2/3) Hyperbolic Matrix(9,2,-40,-9) (-1/4,-1/5) -> (-1/4,-1/5) Reflection Matrix(27,4,20,3) (-1/5,0/1) -> (4/3,7/5) Hyperbolic Matrix(67,-12,28,-5) (0/1,1/5) -> (7/3,12/5) Hyperbolic Matrix(7,-2,24,-7) (1/4,1/3) -> (1/4,1/3) Reflection Matrix(5,-2,12,-5) (1/3,1/2) -> (1/3,1/2) Reflection Matrix(11,-6,20,-11) (1/2,3/5) -> (1/2,3/5) 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(9,-8,8,-7) (3/4,1/1) -> (1/1,5/4) Parabolic Matrix(71,-90,56,-71) (5/4,9/7) -> (5/4,9/7) Reflection Matrix(135,-176,56,-73) (9/7,4/3) -> (12/5,17/7) Hyperbolic Matrix(29,-42,20,-29) (7/5,3/2) -> (7/5,3/2) Reflection Matrix(19,-30,12,-19) (3/2,5/3) -> (3/2,5/3) Reflection Matrix(41,-70,24,-41) (5/3,7/4) -> (5/3,7/4) Reflection Matrix(69,-170,28,-69) (17/7,5/2) -> (17/7,5/2) Reflection Matrix(11,-30,4,-11) (5/2,3/1) -> (5/2,3/1) Reflection Matrix(-1,6,0,1) (3/1,1/0) -> (3/1,1/0) 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(3,2,-4,-3) -> Matrix(1,0,4,-1) (-1/1,-1/2) -> (0/1,1/2) Matrix(13,6,-28,-13) -> Matrix(1,0,4,-1) (-1/2,-3/7) -> (0/1,1/2) Matrix(39,16,56,23) -> Matrix(3,-2,2,-1) 1/1 Matrix(51,20,28,11) -> Matrix(3,-2,8,-5) 1/2 Matrix(11,4,52,19) -> Matrix(1,0,0,1) Matrix(55,16,24,7) -> Matrix(1,-2,2,-3) 1/1 Matrix(43,12,68,19) -> Matrix(1,0,0,1) Matrix(9,2,-40,-9) -> Matrix(1,0,0,-1) (-1/4,-1/5) -> (0/1,1/0) Matrix(27,4,20,3) -> Matrix(1,0,-2,1) 0/1 Matrix(67,-12,28,-5) -> Matrix(1,0,0,1) Matrix(7,-2,24,-7) -> Matrix(5,-4,6,-5) (1/4,1/3) -> (2/3,1/1) Matrix(5,-2,12,-5) -> Matrix(-1,2,0,1) (1/3,1/2) -> (1/1,1/0) Matrix(11,-6,20,-11) -> Matrix(-1,2,0,1) (1/2,3/5) -> (1/1,1/0) Matrix(49,-30,80,-49) -> Matrix(1,0,0,-1) (3/5,5/8) -> (0/1,1/0) Matrix(41,-30,56,-41) -> Matrix(1,2,0,-1) (5/7,3/4) -> (-1/1,1/0) Matrix(9,-8,8,-7) -> Matrix(1,0,2,1) 0/1 Matrix(71,-90,56,-71) -> Matrix(3,-2,4,-3) (5/4,9/7) -> (1/2,1/1) Matrix(135,-176,56,-73) -> Matrix(1,-2,2,-3) 1/1 Matrix(29,-42,20,-29) -> Matrix(1,0,4,-1) (7/5,3/2) -> (0/1,1/2) Matrix(19,-30,12,-19) -> Matrix(1,0,4,-1) (3/2,5/3) -> (0/1,1/2) Matrix(41,-70,24,-41) -> Matrix(1,0,6,-1) (5/3,7/4) -> (0/1,1/3) Matrix(69,-170,28,-69) -> Matrix(3,-2,4,-3) (17/7,5/2) -> (1/2,1/1) Matrix(11,-30,4,-11) -> Matrix(3,-2,4,-3) (5/2,3/1) -> (1/2,1/1) Matrix(-1,6,0,1) -> Matrix(1,0,2,-1) (3/1,1/0) -> (0/1,1/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.