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): 144 Minimal number of generators: 25 Number of equivalence classes of cusps: 20 Genus: 3 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -3/5 -5/9 -1/2 -9/20 -1/4 -1/5 0/1 1/5 1/4 3/10 1/3 2/5 3/7 1/2 3/5 2/3 7/10 4/5 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 1/0 -4/5 0/1 -7/9 0/1 1/1 -3/4 0/1 1/1 1/0 -8/11 0/1 1/1 -5/7 1/1 1/0 -7/10 1/0 -2/3 0/1 1/0 -3/5 1/0 -4/7 -2/1 1/0 -5/9 -2/1 -1/1 -1/2 -1/1 0/1 1/0 -5/11 -2/1 -1/1 -9/20 -1/1 -4/9 -1/1 0/1 -3/7 -1/1 1/0 -2/5 -1/1 -1/3 -1/1 0/1 -3/10 0/1 -2/7 0/1 1/0 -3/11 -1/1 0/1 -1/4 -1/1 0/1 1/0 -1/5 -1/2 1/0 -1/6 -1/1 0/1 1/0 0/1 -1/1 0/1 1/5 -1/2 1/0 2/9 -1/1 0/1 1/4 -1/1 -1/2 0/1 3/11 -1/1 0/1 2/7 -1/2 0/1 3/10 0/1 1/3 -1/1 0/1 2/5 -1/1 3/7 -1/1 -1/2 4/9 -1/1 0/1 1/2 -1/1 -1/2 0/1 3/5 -1/2 5/8 -1/2 -2/5 -1/3 12/19 -2/5 -1/3 19/30 -1/3 7/11 -1/3 0/1 2/3 -1/2 0/1 7/10 -1/2 5/7 -1/2 -1/3 8/11 -1/3 0/1 3/4 -1/2 -1/3 0/1 7/9 -1/3 0/1 4/5 0/1 1/1 -1/2 0/1 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(9,8,10,9) (-1/1,-4/5) -> (4/5,1/1) Hyperbolic Matrix(71,56,90,71) (-4/5,-7/9) -> (7/9,4/5) Hyperbolic Matrix(21,16,80,61) (-7/9,-3/4) -> (1/4,3/11) Hyperbolic Matrix(19,14,80,59) (-3/4,-8/11) -> (2/9,1/4) Hyperbolic Matrix(61,44,140,101) (-8/11,-5/7) -> (3/7,4/9) Hyperbolic Matrix(99,70,140,99) (-5/7,-7/10) -> (7/10,5/7) Hyperbolic Matrix(41,28,60,41) (-7/10,-2/3) -> (2/3,7/10) Hyperbolic Matrix(29,18,-50,-31) (-2/3,-3/5) -> (-3/5,-4/7) Parabolic Matrix(39,22,140,79) (-4/7,-5/9) -> (3/11,2/7) Hyperbolic Matrix(19,10,-40,-21) (-5/9,-1/2) -> (-1/2,-5/11) Parabolic Matrix(349,158,550,249) (-5/11,-9/20) -> (19/30,7/11) Hyperbolic Matrix(411,184,650,291) (-9/20,-4/9) -> (12/19,19/30) Hyperbolic Matrix(101,44,140,61) (-4/9,-3/7) -> (5/7,8/11) Hyperbolic Matrix(29,12,70,29) (-3/7,-2/5) -> (2/5,3/7) Hyperbolic Matrix(11,4,30,11) (-2/5,-1/3) -> (1/3,2/5) Hyperbolic Matrix(19,6,60,19) (-1/3,-3/10) -> (3/10,1/3) Hyperbolic Matrix(41,12,140,41) (-3/10,-2/7) -> (2/7,3/10) Hyperbolic Matrix(71,20,110,31) (-2/7,-3/11) -> (7/11,2/3) Hyperbolic Matrix(61,16,80,21) (-3/11,-1/4) -> (3/4,7/9) Hyperbolic Matrix(9,2,-50,-11) (-1/4,-1/5) -> (-1/5,-1/6) Parabolic Matrix(51,8,70,11) (-1/6,0/1) -> (8/11,3/4) Hyperbolic Matrix(11,-2,50,-9) (0/1,1/5) -> (1/5,2/9) Parabolic Matrix(69,-32,110,-51) (4/9,1/2) -> (5/8,12/19) Hyperbolic Matrix(31,-18,50,-29) (1/2,3/5) -> (3/5,5/8) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-2,1) Matrix(9,8,10,9) -> Matrix(1,0,-2,1) Matrix(71,56,90,71) -> Matrix(1,0,-4,1) Matrix(21,16,80,61) -> Matrix(1,0,-2,1) Matrix(19,14,80,59) -> Matrix(1,0,-2,1) Matrix(61,44,140,101) -> Matrix(1,0,-2,1) Matrix(99,70,140,99) -> Matrix(1,-2,-2,5) Matrix(41,28,60,41) -> Matrix(1,0,-2,1) Matrix(29,18,-50,-31) -> Matrix(1,-2,0,1) Matrix(39,22,140,79) -> Matrix(1,2,-2,-3) Matrix(19,10,-40,-21) -> Matrix(1,0,0,1) Matrix(349,158,550,249) -> Matrix(1,2,-4,-7) Matrix(411,184,650,291) -> Matrix(3,2,-8,-5) Matrix(101,44,140,61) -> Matrix(1,0,-2,1) Matrix(29,12,70,29) -> Matrix(1,2,-2,-3) Matrix(11,4,30,11) -> Matrix(1,0,0,1) Matrix(19,6,60,19) -> Matrix(1,0,0,1) Matrix(41,12,140,41) -> Matrix(1,0,-2,1) Matrix(71,20,110,31) -> Matrix(1,0,-2,1) Matrix(61,16,80,21) -> Matrix(1,0,-2,1) Matrix(9,2,-50,-11) -> Matrix(1,0,0,1) Matrix(51,8,70,11) -> Matrix(1,0,-2,1) Matrix(11,-2,50,-9) -> Matrix(1,0,0,1) Matrix(69,-32,110,-51) -> Matrix(3,2,-8,-5) Matrix(31,-18,50,-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): 3 Degree of the the map X: 3 Degree of the the map Y: 24 Permutation triple for Y: ((1,4,14,21,23,24,18,15,5,2)(3,10,20,8,7,19,13,17,16,11)(6,12)(9,22); (1,2,8,9,3)(4,6,5,17,13)(10,18,12,21,20)(16,22,19,24,23); (2,6,18,19,7)(3,11,23,12,4)(5,15,10,9,16)(8,21,14,13,22)) ----------------------------------------------------------------------- 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 0/1 (-1/1,0/1) 0 10 1/5 0 2 2/9 (-1/1,0/1) 0 10 1/4 0 5 3/11 (-1/1,0/1) 0 10 2/7 (-1/2,0/1) 0 10 3/10 0/1 1 1 1/3 (-1/1,0/1) 0 10 2/5 -1/1 1 2 3/7 (-1/1,-1/2) 0 10 4/9 (-1/1,0/1) 0 10 1/2 0 5 3/5 -1/2 2 2 5/8 0 5 12/19 (-2/5,-1/3) 0 10 19/30 -1/3 2 1 7/11 (-1/3,0/1) 0 10 2/3 (-1/2,0/1) 0 10 7/10 -1/2 1 1 5/7 (-1/2,-1/3) 0 10 8/11 (-1/3,0/1) 0 10 3/4 0 5 7/9 (-1/3,0/1) 0 10 4/5 0/1 1 2 1/1 (-1/2,0/1) 0 10 1/0 0/1 1 1 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(11,-2,50,-9) (0/1,1/5) -> (1/5,2/9) Parabolic Matrix(59,-14,80,-19) (2/9,1/4) -> (8/11,3/4) Glide Reflection Matrix(61,-16,80,-21) (1/4,3/11) -> (3/4,7/9) Glide Reflection Matrix(71,-20,110,-31) (3/11,2/7) -> (7/11,2/3) Glide Reflection Matrix(41,-12,140,-41) (2/7,3/10) -> (2/7,3/10) Reflection Matrix(19,-6,60,-19) (3/10,1/3) -> (3/10,1/3) Reflection Matrix(11,-4,30,-11) (1/3,2/5) -> (1/3,2/5) Reflection Matrix(29,-12,70,-29) (2/5,3/7) -> (2/5,3/7) Reflection Matrix(101,-44,140,-61) (3/7,4/9) -> (5/7,8/11) Glide Reflection Matrix(69,-32,110,-51) (4/9,1/2) -> (5/8,12/19) Hyperbolic Matrix(31,-18,50,-29) (1/2,3/5) -> (3/5,5/8) Parabolic Matrix(721,-456,1140,-721) (12/19,19/30) -> (12/19,19/30) Reflection Matrix(419,-266,660,-419) (19/30,7/11) -> (19/30,7/11) Reflection Matrix(41,-28,60,-41) (2/3,7/10) -> (2/3,7/10) Reflection Matrix(99,-70,140,-99) (7/10,5/7) -> (7/10,5/7) Reflection Matrix(71,-56,90,-71) (7/9,4/5) -> (7/9,4/5) Reflection Matrix(9,-8,10,-9) (4/5,1/1) -> (4/5,1/1) 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(1,0,0,-1) -> Matrix(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) Matrix(11,-2,50,-9) -> Matrix(1,0,0,1) Matrix(59,-14,80,-19) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(61,-16,80,-21) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(71,-20,110,-31) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(41,-12,140,-41) -> Matrix(-1,0,4,1) (2/7,3/10) -> (-1/2,0/1) Matrix(19,-6,60,-19) -> Matrix(-1,0,2,1) (3/10,1/3) -> (-1/1,0/1) Matrix(11,-4,30,-11) -> Matrix(-1,0,2,1) (1/3,2/5) -> (-1/1,0/1) Matrix(29,-12,70,-29) -> Matrix(3,2,-4,-3) (2/5,3/7) -> (-1/1,-1/2) Matrix(101,-44,140,-61) -> Matrix(-1,0,4,1) *** -> (-1/2,0/1) Matrix(69,-32,110,-51) -> Matrix(3,2,-8,-5) -1/2 Matrix(31,-18,50,-29) -> Matrix(3,2,-8,-5) -1/2 Matrix(721,-456,1140,-721) -> Matrix(11,4,-30,-11) (12/19,19/30) -> (-2/5,-1/3) Matrix(419,-266,660,-419) -> Matrix(-1,0,6,1) (19/30,7/11) -> (-1/3,0/1) Matrix(41,-28,60,-41) -> Matrix(-1,0,4,1) (2/3,7/10) -> (-1/2,0/1) Matrix(99,-70,140,-99) -> Matrix(5,2,-12,-5) (7/10,5/7) -> (-1/2,-1/3) Matrix(71,-56,90,-71) -> Matrix(-1,0,6,1) (7/9,4/5) -> (-1/3,0/1) Matrix(9,-8,10,-9) -> Matrix(-1,0,4,1) (4/5,1/1) -> (-1/2,0/1) 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.