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): 48 Minimal number of generators: 9 Number of equivalence classes of cusps: 8 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/3 1/2 2/3 1/1 4/3 2/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -2/1 -1/1 -3/2 -1/1 -4/3 -1/1 -1/2 -1/1 -1/2 -2/3 -1/2 -1/2 -1/3 -2/5 -1/3 -1/3 -1/4 0/1 0/1 1/3 1/2 1/2 1/1 2/3 1/0 1/1 1/0 4/3 -1/1 1/0 3/2 -1/1 5/3 1/0 2/1 -1/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,4,0,1) (-2/1,1/0) -> (2/1,1/0) Parabolic Matrix(5,8,-12,-19) (-2/1,-3/2) -> (-1/2,-2/5) Hyperbolic Matrix(17,24,12,17) (-3/2,-4/3) -> (4/3,3/2) Hyperbolic Matrix(7,8,6,7) (-4/3,-1/1) -> (1/1,4/3) Hyperbolic Matrix(5,4,6,5) (-1/1,-2/3) -> (2/3,1/1) Hyperbolic Matrix(7,4,12,7) (-2/3,-1/2) -> (1/2,2/3) Hyperbolic Matrix(31,12,18,7) (-2/5,-1/3) -> (5/3,2/1) Hyperbolic Matrix(1,0,6,1) (-1/3,0/1) -> (0/1,1/3) Parabolic Matrix(19,-8,12,-5) (1/3,1/2) -> (3/2,5/3) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,4,0,1) -> Matrix(1,0,0,1) Matrix(5,8,-12,-19) -> Matrix(1,0,-2,1) Matrix(17,24,12,17) -> Matrix(3,2,-2,-1) Matrix(7,8,6,7) -> Matrix(3,2,-2,-1) Matrix(5,4,6,5) -> Matrix(1,0,2,1) Matrix(7,4,12,7) -> Matrix(5,2,2,1) Matrix(31,12,18,7) -> Matrix(7,2,-4,-1) Matrix(1,0,6,1) -> Matrix(1,0,6,1) Matrix(19,-8,12,-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): 3 Degree of the the map X: 3 Degree of the the map Y: 8 Permutation triple for Y: ((1,6,2)(3,7,4); (1,4,8,3,2,5)(6,7); (1,3)(2,7,8,4,6,5)) ----------------------------------------------------------------------- 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 3 3 1/3 1/2 3 6 1/2 1/1 2 6 2/3 1/0 3 3 1/1 1/0 1 6 4/3 (-1/1,1/0) 0 3 3/2 -1/1 2 6 5/3 1/0 3 6 2/1 -1/1 1 3 1/0 (-1/1,0/1) 0 6 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(1,0,6,-1) (0/1,1/3) -> (0/1,1/3) Reflection Matrix(19,-8,12,-5) (1/3,1/2) -> (3/2,5/3) Hyperbolic Matrix(7,-4,12,-7) (1/2,2/3) -> (1/2,2/3) Reflection Matrix(5,-4,6,-5) (2/3,1/1) -> (2/3,1/1) Reflection Matrix(7,-8,6,-7) (1/1,4/3) -> (1/1,4/3) Reflection Matrix(17,-24,12,-17) (4/3,3/2) -> (4/3,3/2) Reflection Matrix(11,-20,6,-11) (5/3,2/1) -> (5/3,2/1) Reflection Matrix(-1,4,0,1) (2/1,1/0) -> (2/1,1/0) 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(1,0,6,-1) -> Matrix(1,0,4,-1) (0/1,1/3) -> (0/1,1/2) Matrix(19,-8,12,-5) -> Matrix(1,0,-2,1) 0/1 Matrix(7,-4,12,-7) -> Matrix(-1,2,0,1) (1/2,2/3) -> (1/1,1/0) Matrix(5,-4,6,-5) -> Matrix(1,0,0,-1) (2/3,1/1) -> (0/1,1/0) Matrix(7,-8,6,-7) -> Matrix(1,2,0,-1) (1/1,4/3) -> (-1/1,1/0) Matrix(17,-24,12,-17) -> Matrix(1,2,0,-1) (4/3,3/2) -> (-1/1,1/0) Matrix(11,-20,6,-11) -> Matrix(1,2,0,-1) (5/3,2/1) -> (-1/1,1/0) Matrix(-1,4,0,1) -> Matrix(-1,0,2,1) (2/1,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.