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): 96 Minimal number of generators: 17 Number of equivalence classes of cusps: 16 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -4/1 -3/1 -2/1 -3/2 0/1 1/1 6/5 3/2 2/1 12/5 5/2 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 -6/1 1/0 -5/1 -3/1 -2/1 1/0 -4/1 -2/1 1/0 -3/1 -2/1 -8/3 -2/1 -3/2 -5/2 -3/2 -2/1 -2/1 -1/1 -7/4 -3/2 -12/7 -1/1 -5/3 -2/1 -3/2 -1/1 -3/2 -1/1 -7/5 -1/1 -2/3 -1/2 -4/3 -1/1 0/1 -5/4 1/0 -6/5 1/0 -1/1 -2/1 -1/1 1/0 0/1 -1/1 1/1 -1/1 -1/2 0/1 6/5 -1/2 5/4 -1/2 4/3 -1/2 0/1 3/2 0/1 8/5 0/1 1/0 5/3 -1/1 0/1 1/0 2/1 -1/1 0/1 7/3 -1/1 0/1 1/0 12/5 -1/1 5/2 -1/2 3/1 0/1 7/2 1/2 4/1 0/1 1/1 5/1 1/1 2/1 1/0 6/1 1/0 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,12,0,1) (-6/1,1/0) -> (6/1,1/0) Parabolic Matrix(11,60,2,11) (-6/1,-5/1) -> (5/1,6/1) Hyperbolic Matrix(13,60,8,37) (-5/1,-4/1) -> (8/5,5/3) Hyperbolic Matrix(11,36,-4,-13) (-4/1,-3/1) -> (-3/1,-8/3) Parabolic Matrix(23,60,18,47) (-8/3,-5/2) -> (5/4,4/3) Hyperbolic Matrix(11,24,-6,-13) (-5/2,-2/1) -> (-2/1,-7/4) Parabolic Matrix(83,144,34,59) (-7/4,-12/7) -> (12/5,5/2) Hyperbolic Matrix(85,144,36,61) (-12/7,-5/3) -> (7/3,12/5) Hyperbolic Matrix(23,36,-16,-25) (-5/3,-3/2) -> (-3/2,-7/5) Parabolic Matrix(35,48,8,11) (-7/5,-4/3) -> (4/1,5/1) Hyperbolic Matrix(37,48,10,13) (-4/3,-5/4) -> (7/2,4/1) Hyperbolic Matrix(49,60,40,49) (-5/4,-6/5) -> (6/5,5/4) Hyperbolic Matrix(11,12,10,11) (-6/5,-1/1) -> (1/1,6/5) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(13,-24,6,-11) (5/3,2/1) -> (2/1,7/3) Parabolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,12,0,1) -> Matrix(1,2,0,1) Matrix(11,60,2,11) -> Matrix(1,4,0,1) Matrix(13,60,8,37) -> Matrix(1,2,0,1) Matrix(11,36,-4,-13) -> Matrix(3,8,-2,-5) Matrix(23,60,18,47) -> Matrix(1,2,-4,-7) Matrix(11,24,-6,-13) -> Matrix(1,0,0,1) Matrix(83,144,34,59) -> Matrix(3,4,-4,-5) Matrix(85,144,36,61) -> Matrix(1,2,-2,-3) Matrix(23,36,-16,-25) -> Matrix(3,4,-4,-5) Matrix(35,48,8,11) -> Matrix(1,0,2,1) Matrix(37,48,10,13) -> Matrix(1,0,2,1) Matrix(49,60,40,49) -> Matrix(1,0,-2,1) Matrix(11,12,10,11) -> Matrix(1,2,-2,-3) Matrix(1,0,2,1) -> Matrix(1,2,-2,-3) Matrix(25,-36,16,-23) -> Matrix(1,0,2,1) Matrix(13,-24,6,-11) -> Matrix(1,0,0,1) Matrix(13,-36,4,-11) -> Matrix(1,0,4,1) 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): 4 Degree of the the map X: 4 Degree of the the map Y: 16 Permutation triple for Y: ((2,6,7)(3,11,4)(5,10,9)(8,13,12); (1,4,13,14,5,2)(3,10)(6,9,16,12,11,15)(7,8); (1,2,8,16,9,3)(4,12)(5,6)(7,15,11,10,14,13)) ----------------------------------------------------------------------- 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 -4/1 (-2/1,1/0) 0 6 -3/1 -2/1 1 2 -2/1 (-2/1,-1/1) 0 6 -3/2 -1/1 2 2 -4/3 (-1/1,0/1) 0 6 -1/1 0 6 0/1 -1/1 2 2 1/1 0 6 6/5 -1/2 2 2 5/4 -1/2 1 6 4/3 (-1/2,0/1) 0 6 3/2 0/1 1 2 2/1 (-1/1,0/1) 0 6 3/1 0/1 2 2 4/1 (0/1,1/1) 0 6 5/1 0 6 6/1 1/0 2 2 1/0 1/0 1 6 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,24,4,19) (-4/1,1/0) -> (5/4,4/3) Glide Reflection Matrix(7,24,-2,-7) (-4/1,-3/1) -> (-4/1,-3/1) Reflection Matrix(5,12,-2,-5) (-3/1,-2/1) -> (-3/1,-2/1) Reflection Matrix(7,12,-4,-7) (-2/1,-3/2) -> (-2/1,-3/2) Reflection Matrix(17,24,-12,-17) (-3/2,-4/3) -> (-3/2,-4/3) Reflection Matrix(19,24,4,5) (-4/3,-1/1) -> (4/1,5/1) Glide Reflection Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(31,-36,6,-7) (1/1,6/5) -> (5/1,6/1) Glide Reflection Matrix(29,-36,4,-5) (6/5,5/4) -> (6/1,1/0) Glide Reflection Matrix(17,-24,12,-17) (4/3,3/2) -> (4/3,3/2) Reflection Matrix(7,-12,4,-7) (3/2,2/1) -> (3/2,2/1) Reflection Matrix(5,-12,2,-5) (2/1,3/1) -> (2/1,3/1) Reflection Matrix(7,-24,2,-7) (3/1,4/1) -> (3/1,4/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(5,24,4,19) -> Matrix(1,2,-2,-5) Matrix(7,24,-2,-7) -> Matrix(1,4,0,-1) (-4/1,-3/1) -> (-2/1,1/0) Matrix(5,12,-2,-5) -> Matrix(3,4,-2,-3) (-3/1,-2/1) -> (-2/1,-1/1) Matrix(7,12,-4,-7) -> Matrix(3,4,-2,-3) (-2/1,-3/2) -> (-2/1,-1/1) Matrix(17,24,-12,-17) -> Matrix(-1,0,2,1) (-3/2,-4/3) -> (-1/1,0/1) Matrix(19,24,4,5) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(1,0,2,1) -> Matrix(1,2,-2,-3) -1/1 Matrix(31,-36,6,-7) -> Matrix(3,2,2,1) Matrix(29,-36,4,-5) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(17,-24,12,-17) -> Matrix(-1,0,4,1) (4/3,3/2) -> (-1/2,0/1) Matrix(7,-12,4,-7) -> Matrix(-1,0,2,1) (3/2,2/1) -> (-1/1,0/1) Matrix(5,-12,2,-5) -> Matrix(-1,0,2,1) (2/1,3/1) -> (-1/1,0/1) Matrix(7,-24,2,-7) -> Matrix(1,0,2,-1) (3/1,4/1) -> (0/1,1/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.