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 -3/1 -2/1 -3/2 -1/1 -3/5 0/1 1/2 3/5 1/1 3/2 5/3 2/1 3/1 5/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -3/1 -1/2 1/0 -5/2 0/1 -12/5 -1/1 1/1 -7/3 0/1 1/0 -2/1 -1/1 -7/4 0/1 -12/7 -1/1 1/1 -5/3 1/0 -3/2 -1/1 -1/1 -1/1 0/1 -3/4 -1/1 -5/7 -1/2 -2/3 -1/1 -3/5 -1/2 1/0 -4/7 -1/1 -1/2 0/1 0/1 -1/1 1/2 -2/3 3/5 -2/3 5/8 -2/3 2/3 -3/5 1/1 -1/2 4/3 -1/1 3/2 -1/2 8/5 -3/7 5/3 -1/2 -2/5 12/7 -3/7 -1/3 7/4 -2/5 2/1 -1/3 3/1 0/1 4/1 -1/1 5/1 -1/2 6/1 -1/1 -1/3 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,18,-2,-7) (-3/1,1/0) -> (-3/1,-5/2) Parabolic Matrix(17,42,2,5) (-5/2,-12/5) -> (6/1,1/0) Hyperbolic Matrix(61,144,36,85) (-12/5,-7/3) -> (5/3,12/7) Hyperbolic Matrix(29,66,18,41) (-7/3,-2/1) -> (8/5,5/3) Hyperbolic Matrix(13,24,20,37) (-2/1,-7/4) -> (5/8,2/3) Hyperbolic Matrix(97,168,56,97) (-7/4,-12/7) -> (12/7,7/4) Hyperbolic Matrix(53,90,10,17) (-12/7,-5/3) -> (5/1,6/1) Hyperbolic Matrix(19,30,-26,-41) (-5/3,-3/2) -> (-3/4,-5/7) Hyperbolic Matrix(5,6,-6,-7) (-3/2,-1/1) -> (-1/1,-3/4) Parabolic Matrix(43,30,10,7) (-5/7,-2/3) -> (4/1,5/1) Hyperbolic Matrix(29,18,-50,-31) (-2/3,-3/5) -> (-3/5,-4/7) Parabolic Matrix(53,30,30,17) (-4/7,-1/2) -> (7/4,2/1) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(31,-18,50,-29) (1/2,3/5) -> (3/5,5/8) Parabolic Matrix(7,-6,6,-5) (2/3,1/1) -> (1/1,4/3) Parabolic Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(7,-18,2,-5) (2/1,3/1) -> (3/1,4/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(5,18,-2,-7) -> Matrix(1,0,0,1) Matrix(17,42,2,5) -> Matrix(1,0,-2,1) Matrix(61,144,36,85) -> Matrix(1,-2,-2,5) Matrix(29,66,18,41) -> Matrix(1,-2,-2,5) Matrix(13,24,20,37) -> Matrix(5,2,-8,-3) Matrix(97,168,56,97) -> Matrix(1,-2,-2,5) Matrix(53,90,10,17) -> Matrix(1,0,-2,1) Matrix(19,30,-26,-41) -> Matrix(1,2,-2,-3) Matrix(5,6,-6,-7) -> Matrix(1,0,0,1) Matrix(43,30,10,7) -> Matrix(1,0,0,1) Matrix(29,18,-50,-31) -> Matrix(1,0,0,1) Matrix(53,30,30,17) -> Matrix(3,2,-8,-5) Matrix(1,0,4,1) -> Matrix(1,2,-2,-3) Matrix(31,-18,50,-29) -> Matrix(11,8,-18,-13) Matrix(7,-6,6,-5) -> Matrix(3,2,-8,-5) Matrix(25,-36,16,-23) -> Matrix(7,4,-16,-9) Matrix(7,-18,2,-5) -> Matrix(1,0,2,1) 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): 4 Degree of the the map X: 4 Degree of the the map Y: 16 Permutation triple for Y: ((1,2)(3,9,14,6,13,10)(4,11,8,7,12,5)(15,16); (1,5,6)(2,8,3)(9,11,15)(12,16,13); (1,3,10,16,11,4)(2,6,14,15,12,7)(5,13)(8,9)) ----------------------------------------------------------------------- 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 -3/1 0 2 -5/2 0/1 1 6 -2/1 -1/1 1 6 -5/3 1/0 2 6 -3/2 -1/1 1 2 -1/1 (-1/1,0/1) 0 6 0/1 -1/1 1 2 1/1 -1/2 2 6 3/2 -1/2 2 2 5/3 (-1/2,-2/5) 0 6 12/7 (-1/2,-2/5) 0 2 7/4 -2/5 1 6 2/1 -1/3 1 6 3/1 0/1 2 2 4/1 -1/1 1 6 5/1 -1/2 2 6 6/1 (-1/2,0/1) 0 2 1/0 0/1 1 6 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,18,-2,-7) (-3/1,1/0) -> (-3/1,-5/2) Parabolic Matrix(11,24,6,13) (-5/2,-2/1) -> (7/4,2/1) Glide Reflection Matrix(17,30,4,7) (-2/1,-5/3) -> (4/1,5/1) Glide Reflection Matrix(19,30,-12,-19) (-5/3,-3/2) -> (-5/3,-3/2) Reflection Matrix(5,6,-4,-5) (-3/2,-1/1) -> (-3/2,-1/1) Reflection Matrix(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Reflection Matrix(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(5,-6,4,-5) (1/1,3/2) -> (1/1,3/2) Reflection Matrix(19,-30,12,-19) (3/2,5/3) -> (3/2,5/3) Reflection Matrix(71,-120,42,-71) (5/3,12/7) -> (5/3,12/7) Reflection Matrix(31,-54,4,-7) (12/7,7/4) -> (6/1,1/0) Glide Reflection Matrix(7,-18,2,-5) (2/1,3/1) -> (3/1,4/1) Parabolic Matrix(11,-60,2,-11) (5/1,6/1) -> (5/1,6/1) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(5,18,-2,-7) -> Matrix(1,0,0,1) Matrix(11,24,6,13) -> Matrix(1,2,-2,-5) Matrix(17,30,4,7) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(19,30,-12,-19) -> Matrix(1,2,0,-1) (-5/3,-3/2) -> (-1/1,1/0) Matrix(5,6,-4,-5) -> Matrix(-1,0,2,1) (-3/2,-1/1) -> (-1/1,0/1) Matrix(-1,0,2,1) -> Matrix(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Matrix(1,0,2,-1) -> Matrix(3,2,-4,-3) (0/1,1/1) -> (-1/1,-1/2) Matrix(5,-6,4,-5) -> Matrix(-1,0,4,1) (1/1,3/2) -> (-1/2,0/1) Matrix(19,-30,12,-19) -> Matrix(9,4,-20,-9) (3/2,5/3) -> (-1/2,-2/5) Matrix(71,-120,42,-71) -> Matrix(9,4,-20,-9) (5/3,12/7) -> (-1/2,-2/5) Matrix(31,-54,4,-7) -> Matrix(5,2,-12,-5) *** -> (-1/2,-1/3) Matrix(7,-18,2,-5) -> Matrix(1,0,2,1) 0/1 Matrix(11,-60,2,-11) -> Matrix(-1,0,4,1) (5/1,6/1) -> (-1/2,0/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.