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: 14 Genus: 2 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/2 -7/16 0/1 1/5 1/4 1/3 3/8 3/7 1/2 5/8 2/3 3/4 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 -3/4 0/1 -5/7 0/1 1/1 -2/3 -1/1 -5/8 0/1 -3/5 0/1 1/2 -4/7 1/1 -1/2 -1/1 1/1 -4/9 1/1 -7/16 1/1 -3/7 1/1 2/1 -2/5 3/1 -3/8 1/0 -1/3 -1/1 1/0 -1/4 1/0 -1/5 -2/1 1/0 0/1 -1/1 1/5 -2/3 -1/2 1/4 -1/2 1/3 -1/1 -1/2 3/8 -1/2 2/5 -3/7 3/7 -2/5 -1/3 1/2 -1/1 -1/3 5/9 -2/5 -1/3 9/16 -1/3 4/7 -1/3 3/5 -1/4 0/1 5/8 0/1 2/3 -1/1 5/7 -1/3 0/1 3/4 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(7,6,8,7) (-1/1,-3/4) -> (3/4,1/1) Hyperbolic Matrix(41,30,56,41) (-3/4,-5/7) -> (5/7,3/4) Hyperbolic Matrix(23,16,56,39) (-5/7,-2/3) -> (2/5,3/7) Hyperbolic Matrix(31,20,48,31) (-2/3,-5/8) -> (5/8,2/3) Hyperbolic Matrix(49,30,80,49) (-5/8,-3/5) -> (3/5,5/8) Hyperbolic Matrix(7,4,40,23) (-3/5,-4/7) -> (0/1,1/5) Hyperbolic Matrix(15,8,-32,-17) (-4/7,-1/2) -> (-1/2,-4/9) Parabolic Matrix(145,64,256,113) (-4/9,-7/16) -> (9/16,4/7) Hyperbolic Matrix(143,62,256,111) (-7/16,-3/7) -> (5/9,9/16) Hyperbolic Matrix(39,16,56,23) (-3/7,-2/5) -> (2/3,5/7) Hyperbolic Matrix(31,12,80,31) (-2/5,-3/8) -> (3/8,2/5) Hyperbolic Matrix(17,6,48,17) (-3/8,-1/3) -> (1/3,3/8) Hyperbolic Matrix(7,2,24,7) (-1/3,-1/4) -> (1/4,1/3) Hyperbolic Matrix(9,2,40,9) (-1/4,-1/5) -> (1/5,1/4) Hyperbolic Matrix(23,4,40,7) (-1/5,0/1) -> (4/7,3/5) Hyperbolic Matrix(17,-8,32,-15) (3/7,1/2) -> (1/2,5/9) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-2,1) Matrix(7,6,8,7) -> Matrix(1,0,-2,1) Matrix(41,30,56,41) -> Matrix(1,0,-4,1) Matrix(23,16,56,39) -> Matrix(1,-2,-2,5) Matrix(31,20,48,31) -> Matrix(1,0,0,1) Matrix(49,30,80,49) -> Matrix(1,0,-6,1) Matrix(7,4,40,23) -> Matrix(3,-2,-4,3) Matrix(15,8,-32,-17) -> Matrix(1,0,0,1) Matrix(145,64,256,113) -> Matrix(3,-2,-10,7) Matrix(143,62,256,111) -> Matrix(3,-4,-8,11) Matrix(39,16,56,23) -> Matrix(1,-2,-2,5) Matrix(31,12,80,31) -> Matrix(1,-6,-2,13) Matrix(17,6,48,17) -> Matrix(1,2,-2,-3) Matrix(7,2,24,7) -> Matrix(1,2,-2,-3) Matrix(9,2,40,9) -> Matrix(1,4,-2,-7) Matrix(23,4,40,7) -> Matrix(1,2,-4,-7) Matrix(17,-8,32,-15) -> Matrix(1,0,0,1) 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): 5 Degree of the the map X: 5 Degree of the the map Y: 16 Permutation triple for Y: ((1,4,14,15,16,11,5,2)(3,9,6,13,12,8,7,10); (1,2,8,3)(4,13,6,5)(7,14,11,10)(9,12,16,15); (2,6,15,7)(3,11,12,4)(5,14)(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 0/1 -1/1 1 8 1/5 (-2/3,-1/2) 0 8 1/4 -1/2 1 2 1/3 (-1/1,-1/2) 0 8 3/8 -1/2 4 1 2/5 -3/7 1 8 3/7 (-2/5,-1/3) 0 8 1/2 0 4 5/9 (-2/5,-1/3) 0 8 9/16 -1/3 3 1 4/7 -1/3 1 8 3/5 (-1/4,0/1) 0 8 5/8 0/1 3 1 2/3 -1/1 1 8 5/7 (-1/3,0/1) 0 8 3/4 0/1 1 2 1/1 (-1/2,0/1) 0 8 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(23,-4,40,-7) (0/1,1/5) -> (4/7,3/5) Glide Reflection Matrix(9,-2,40,-9) (1/5,1/4) -> (1/5,1/4) Reflection Matrix(7,-2,24,-7) (1/4,1/3) -> (1/4,1/3) Reflection Matrix(17,-6,48,-17) (1/3,3/8) -> (1/3,3/8) Reflection Matrix(31,-12,80,-31) (3/8,2/5) -> (3/8,2/5) Reflection Matrix(39,-16,56,-23) (2/5,3/7) -> (2/3,5/7) Glide Reflection Matrix(17,-8,32,-15) (3/7,1/2) -> (1/2,5/9) Parabolic Matrix(161,-90,288,-161) (5/9,9/16) -> (5/9,9/16) Reflection Matrix(127,-72,224,-127) (9/16,4/7) -> (9/16,4/7) Reflection Matrix(49,-30,80,-49) (3/5,5/8) -> (3/5,5/8) Reflection Matrix(31,-20,48,-31) (5/8,2/3) -> (5/8,2/3) Reflection Matrix(41,-30,56,-41) (5/7,3/4) -> (5/7,3/4) Reflection Matrix(7,-6,8,-7) (3/4,1/1) -> (3/4,1/1) 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(23,-4,40,-7) -> Matrix(3,2,-10,-7) Matrix(9,-2,40,-9) -> Matrix(7,4,-12,-7) (1/5,1/4) -> (-2/3,-1/2) Matrix(7,-2,24,-7) -> Matrix(3,2,-4,-3) (1/4,1/3) -> (-1/1,-1/2) Matrix(17,-6,48,-17) -> Matrix(3,2,-4,-3) (1/3,3/8) -> (-1/1,-1/2) Matrix(31,-12,80,-31) -> Matrix(13,6,-28,-13) (3/8,2/5) -> (-1/2,-3/7) Matrix(39,-16,56,-23) -> Matrix(5,2,-12,-5) *** -> (-1/2,-1/3) Matrix(17,-8,32,-15) -> Matrix(1,0,0,1) Matrix(161,-90,288,-161) -> Matrix(11,4,-30,-11) (5/9,9/16) -> (-2/5,-1/3) Matrix(127,-72,224,-127) -> Matrix(7,2,-24,-7) (9/16,4/7) -> (-1/3,-1/4) Matrix(49,-30,80,-49) -> Matrix(-1,0,8,1) (3/5,5/8) -> (-1/4,0/1) Matrix(31,-20,48,-31) -> Matrix(-1,0,2,1) (5/8,2/3) -> (-1/1,0/1) Matrix(41,-30,56,-41) -> Matrix(-1,0,6,1) (5/7,3/4) -> (-1/3,0/1) Matrix(7,-6,8,-7) -> Matrix(-1,0,4,1) (3/4,1/1) -> (-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 has no extra symmetries.