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 -1/1 -5/1 -3/4 -4/1 -1/1 -1/2 -3/1 -1/1 -8/3 -1/1 -3/4 -5/2 -3/4 -2/3 -2/1 -1/2 -7/4 -1/4 0/1 -12/7 0/1 -5/3 1/0 -3/2 -2/3 0/1 -7/5 1/0 -4/3 -1/1 -1/2 -5/4 -2/3 -1/2 -6/5 -1/2 -1/1 -1/2 0/1 0/1 1/1 1/0 6/5 1/0 5/4 -2/1 1/0 4/3 -1/1 1/0 3/2 -2/1 0/1 8/5 -1/1 1/0 5/3 -1/2 2/1 1/0 7/3 -5/2 12/5 -2/1 5/2 -2/1 -3/2 3/1 -1/1 7/2 0/1 1/0 4/1 -1/1 1/0 5/1 -3/2 6/1 -1/1 1/0 -1/1 0/1 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(1,0,0,1) Matrix(11,60,2,11) -> Matrix(7,6,-6,-5) Matrix(13,60,8,37) -> Matrix(3,2,-2,-1) Matrix(11,36,-4,-13) -> Matrix(1,2,-2,-3) Matrix(23,60,18,47) -> Matrix(5,4,-4,-3) Matrix(11,24,-6,-13) -> Matrix(3,2,-8,-5) Matrix(83,144,34,59) -> Matrix(11,2,-6,-1) Matrix(85,144,36,61) -> Matrix(5,-2,-2,1) Matrix(23,36,-16,-25) -> Matrix(1,0,0,1) Matrix(35,48,8,11) -> Matrix(3,2,-2,-1) Matrix(37,48,10,13) -> Matrix(3,2,-2,-1) Matrix(49,60,40,49) -> Matrix(7,4,-2,-1) Matrix(11,12,10,11) -> Matrix(3,2,-2,-1) Matrix(1,0,2,1) -> Matrix(1,0,2,1) Matrix(25,-36,16,-23) -> Matrix(1,0,0,1) Matrix(13,-24,6,-11) -> Matrix(1,-2,0,1) Matrix(13,-36,4,-11) -> Matrix(1,2,-2,-3) 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): 5 Degree of the the map X: 5 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 0/1 0/1 2 2 1/1 1/0 2 12 6/5 1/0 2 2 5/4 (-2/1,1/0) 0 12 4/3 0 6 3/2 0 4 8/5 0 6 5/3 -1/2 2 12 2/1 1/0 2 6 7/3 -5/2 2 12 12/5 -2/1 8 2 5/2 (-2/1,-3/2) 0 12 3/1 -1/1 2 4 7/2 (0/1,1/0) 0 12 4/1 0 6 5/1 -3/2 2 12 6/1 -1/1 6 2 1/0 (-1/1,0/1) 0 12 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(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(11,-12,10,-11) (1/1,6/5) -> (1/1,6/5) Reflection Matrix(49,-60,40,-49) (6/5,5/4) -> (6/5,5/4) Reflection Matrix(37,-48,10,-13) (5/4,4/3) -> (7/2,4/1) Glide Reflection Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(37,-60,8,-13) (8/5,5/3) -> (4/1,5/1) Glide Reflection Matrix(13,-24,6,-11) (5/3,2/1) -> (2/1,7/3) Parabolic Matrix(71,-168,30,-71) (7/3,12/5) -> (7/3,12/5) Reflection Matrix(49,-120,20,-49) (12/5,5/2) -> (12/5,5/2) Reflection Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic Matrix(11,-60,2,-11) (5/1,6/1) -> (5/1,6/1) Reflection Matrix(-1,12,0,1) (6/1,1/0) -> (6/1,1/0) 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(1,0,2,-1) -> Matrix(1,0,0,-1) (0/1,1/1) -> (0/1,1/0) Matrix(11,-12,10,-11) -> Matrix(1,2,0,-1) (1/1,6/5) -> (-1/1,1/0) Matrix(49,-60,40,-49) -> Matrix(1,4,0,-1) (6/5,5/4) -> (-2/1,1/0) Matrix(37,-48,10,-13) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(25,-36,16,-23) -> Matrix(1,0,0,1) Matrix(37,-60,8,-13) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(13,-24,6,-11) -> Matrix(1,-2,0,1) 1/0 Matrix(71,-168,30,-71) -> Matrix(9,20,-4,-9) (7/3,12/5) -> (-5/2,-2/1) Matrix(49,-120,20,-49) -> Matrix(7,12,-4,-7) (12/5,5/2) -> (-2/1,-3/2) Matrix(13,-36,4,-11) -> Matrix(1,2,-2,-3) -1/1 Matrix(11,-60,2,-11) -> Matrix(5,6,-4,-5) (5/1,6/1) -> (-3/2,-1/1) Matrix(-1,12,0,1) -> Matrix(-1,0,2,1) (6/1,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.