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 -5/1 -3/1 -2/1 -1/1 0/1 1/3 1/2 2/3 1/1 2/1 3/1 4/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 1/0 -4/1 -2/1 1/0 -7/2 -2/1 -3/2 -1/1 -3/1 -1/1 -2/1 -1/1 0/1 -1/1 0/1 -2/3 0/1 1/3 -3/5 1/3 -4/7 2/5 1/2 -1/2 0/1 1/2 1/1 -1/3 1/3 1/1 -1/4 1/2 2/3 1/1 0/1 0/1 1/1 1/4 1/2 2/3 1/1 1/3 1/1 3/8 1/1 3/2 2/1 2/5 1/1 2/1 3/7 1/1 1/2 1/1 2/1 1/0 4/7 4/1 1/0 3/5 1/0 2/3 0/1 1/0 5/7 1/0 3/4 -2/1 -1/1 1/0 1/1 -1/1 1/1 3/2 -1/1 0/1 1/0 2/1 0/1 1/0 5/2 -1/1 0/1 1/0 3/1 0/1 7/2 0/1 1/4 1/3 4/1 0/1 1/2 5/1 1/1 6/1 0/1 1/1 1/0 0/1 1/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(3,20,4,27) (-5/1,1/0) -> (5/7,3/4) Hyperbolic Matrix(7,32,12,55) (-5/1,-4/1) -> (4/7,3/5) Hyperbolic Matrix(15,56,4,15) (-4/1,-7/2) -> (7/2,4/1) Hyperbolic Matrix(7,24,16,55) (-7/2,-3/1) -> (3/7,1/2) Hyperbolic Matrix(5,12,-8,-19) (-3/1,-2/1) -> (-2/3,-3/5) Hyperbolic Matrix(3,4,-4,-5) (-2/1,-1/1) -> (-1/1,-2/3) Parabolic Matrix(55,32,12,7) (-3/5,-4/7) -> (4/1,5/1) Hyperbolic Matrix(15,8,28,15) (-4/7,-1/2) -> (1/2,4/7) Hyperbolic Matrix(11,4,8,3) (-1/2,-1/3) -> (1/1,3/2) Hyperbolic Matrix(13,4,16,5) (-1/3,-1/4) -> (3/4,1/1) Hyperbolic Matrix(1,0,8,1) (-1/4,0/1) -> (0/1,1/4) Parabolic Matrix(13,-4,36,-11) (1/4,1/3) -> (1/3,3/8) Parabolic Matrix(53,-20,8,-3) (3/8,2/5) -> (6/1,1/0) Hyperbolic Matrix(67,-28,12,-5) (2/5,3/7) -> (5/1,6/1) Hyperbolic Matrix(25,-16,36,-23) (3/5,2/3) -> (2/3,5/7) Parabolic Matrix(9,-16,4,-7) (3/2,2/1) -> (2/1,5/2) Parabolic Matrix(13,-36,4,-11) (5/2,3/1) -> (3/1,7/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(3,20,4,27) -> Matrix(1,-2,0,1) Matrix(7,32,12,55) -> Matrix(1,6,0,1) Matrix(15,56,4,15) -> Matrix(1,2,2,5) Matrix(7,24,16,55) -> Matrix(3,4,2,3) Matrix(5,12,-8,-19) -> Matrix(1,0,4,1) Matrix(3,4,-4,-5) -> Matrix(1,0,4,1) Matrix(55,32,12,7) -> Matrix(5,-2,8,-3) Matrix(15,8,28,15) -> Matrix(3,-2,2,-1) Matrix(11,4,8,3) -> Matrix(1,0,-2,1) Matrix(13,4,16,5) -> Matrix(1,0,-2,1) Matrix(1,0,8,1) -> Matrix(1,0,0,1) Matrix(13,-4,36,-11) -> Matrix(5,-4,4,-3) Matrix(53,-20,8,-3) -> Matrix(1,-2,2,-3) Matrix(67,-28,12,-5) -> Matrix(1,-2,2,-3) Matrix(25,-16,36,-23) -> Matrix(1,0,0,1) Matrix(9,-16,4,-7) -> 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: ((1,7,8,2)(3,9)(4,5)(6,13)(10,12,11,14)(15,16); (1,5,8,6)(2,3)(4,12)(7,16)(9,11,15,10)(13,14); (1,3,10,13,8,16,11,4)(2,5,12,15,7,6,14,9)) ----------------------------------------------------------------------- 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 -4/1 (-2/1,1/0) 0 8 -3/1 -1/1 2 4 -2/1 (-1/1,0/1) 0 8 -1/1 0/1 2 2 0/1 (0/1,1/1) 0 8 1/3 1/1 2 2 2/5 (1/1,2/1) 0 8 3/7 1/1 2 4 1/2 0 8 4/7 (4/1,1/0) 0 8 3/5 1/0 4 2 2/3 (0/1,1/0) 0 8 1/1 (0/1,1/0) 0 4 2/1 (0/1,1/0) 0 8 3/1 0/1 2 2 4/1 (0/1,1/2) 0 8 5/1 1/1 2 4 1/0 0 8 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,8,2,15) (-4/1,1/0) -> (1/2,4/7) Glide Reflection Matrix(9,32,2,7) (-4/1,-3/1) -> (4/1,5/1) Glide Reflection Matrix(5,12,-2,-5) (-3/1,-2/1) -> (-3/1,-2/1) Reflection Matrix(3,4,-2,-3) (-2/1,-1/1) -> (-2/1,-1/1) Reflection Matrix(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Reflection Matrix(1,0,6,-1) (0/1,1/3) -> (0/1,1/3) 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(17,-8,2,-1) (3/7,1/2) -> (5/1,1/0) Glide Reflection Matrix(41,-24,70,-41) (4/7,3/5) -> (4/7,3/5) Reflection Matrix(19,-12,30,-19) (3/5,2/3) -> (3/5,2/3) Reflection Matrix(5,-4,6,-5) (2/3,1/1) -> (2/3,1/1) Reflection Matrix(3,-4,2,-3) (1/1,2/1) -> (1/1,2/1) 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(1,8,2,15) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(9,32,2,7) -> Matrix(1,2,2,3) Matrix(5,12,-2,-5) -> Matrix(-1,0,2,1) (-3/1,-2/1) -> (-1/1,0/1) Matrix(3,4,-2,-3) -> Matrix(-1,0,2,1) (-2/1,-1/1) -> (-1/1,0/1) Matrix(-1,0,2,1) -> Matrix(1,0,2,-1) (-1/1,0/1) -> (0/1,1/1) Matrix(1,0,6,-1) -> Matrix(1,0,2,-1) (0/1,1/3) -> (0/1,1/1) Matrix(11,-4,30,-11) -> Matrix(3,-4,2,-3) (1/3,2/5) -> (1/1,2/1) Matrix(29,-12,70,-29) -> Matrix(3,-4,2,-3) (2/5,3/7) -> (1/1,2/1) Matrix(17,-8,2,-1) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(41,-24,70,-41) -> Matrix(-1,8,0,1) (4/7,3/5) -> (4/1,1/0) Matrix(19,-12,30,-19) -> Matrix(1,0,0,-1) (3/5,2/3) -> (0/1,1/0) Matrix(5,-4,6,-5) -> Matrix(1,0,0,-1) (2/3,1/1) -> (0/1,1/0) Matrix(3,-4,2,-3) -> Matrix(1,0,0,-1) (1/1,2/1) -> (0/1,1/0) Matrix(5,-12,2,-5) -> Matrix(1,0,0,-1) (2/1,3/1) -> (0/1,1/0) Matrix(7,-24,2,-7) -> Matrix(1,0,4,-1) (3/1,4/1) -> (0/1,1/2) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.