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 -4/1 -2/1 0/1 1/1 2/1 8/3 4/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(3,16,2,11) (-4/1,1/0) -> (4/3,3/2) Hyperbolic Matrix(5,16,4,13) (-4/1,-3/1) -> (1/1,4/3) Hyperbolic Matrix(7,16,-4,-9) (-3/1,-2/1) -> (-2/1,-5/3) Parabolic Matrix(39,64,14,23) (-5/3,-8/5) -> (8/3,3/1) Hyperbolic Matrix(41,64,16,25) (-8/5,-3/2) -> (5/2,8/3) Hyperbolic Matrix(11,16,2,3) (-3/2,-4/3) -> (4/1,1/0) Hyperbolic Matrix(13,16,4,5) (-4/3,-1/1) -> (3/1,4/1) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(9,-16,4,-7) (3/2,2/1) -> (2/1,5/2) Parabolic Since the preimage of every curve is trivial, the pure modular group virtual endomorphism is trivial. ----------------------------------------------------------------------- Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift elements of DeckMod(f) via pi_1: 0, lambda1 DeckMod(f) is isomorphic to Z/2Z. Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift modular group liftables via pi_1: 0, lambda1 The subgroup of modular group liftables which arise from translations is trivial. ----------------------------------------------------------------------- INFORMATION ON THE IMAGE IN PSL(2,Z) OF THE GROUP OF MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE Index in PSL(2,Z): 24 Minimal number of generators: 5 Number of equivalence classes of elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 6 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 4/3 2/1 8/3 4/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 -1/2 1/0 1/1 -1/1 0/1 4/3 -1/1 0/1 3/2 -1/1 0/1 2/1 -1/1 0/1 5/2 -1/1 0/1 8/3 -1/1 0/1 3/1 -1/1 0/1 4/1 -1/1 0/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,1,1) (0/1,1/0) -> (0/1,1/1) Parabolic Matrix(13,-16,9,-11) (1/1,4/3) -> (4/3,3/2) Parabolic Matrix(9,-16,4,-7) (3/2,2/1) -> (2/1,5/2) Parabolic Matrix(25,-64,9,-23) (5/2,8/3) -> (8/3,3/1) Parabolic Matrix(5,-16,1,-3) (3/1,4/1) -> (4/1,1/0) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL MULTI-ENDOMORPHISM This map is 2-valued. Matrix(1,0,1,1) -> Matrix(1,0,0,1) -> Matrix(1,1,-2,-1) Matrix(13,-16,9,-11) -> Matrix(1,0,0,1) -> Matrix(1,1,-2,-1) Matrix(9,-16,4,-7) -> Matrix(1,0,0,1) -> Matrix(1,1,-2,-1) Matrix(25,-64,9,-23) -> Matrix(1,0,0,1) -> Matrix(1,1,-2,-1) Matrix(5,-16,1,-3) -> Matrix(1,0,0,1) -> Matrix(1,1,-2,-1) Matrix(1,0,0,1) -> Matrix(1,0,0,1) -> Matrix(1,1,-2,-1) ----------------------------------------------------------------------- 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 0/1 (-1/1,0/1).(-1/2,1/0) 2/1 (-1/1,0/1).(-1/2,1/0) 8/3 (-1/1,0/1).(-1/2,1/0) 3/1 (-1/1,0/1).(-1/2,1/0) 4/1 (-1/1,0/1).(-1/2,1/0) 1/0 (-1/1,0/1).(-1/2,1/0) 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(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) Reflection Matrix(7,-16,3,-7) (2/1,8/3) -> (2/1,8/3) Reflection Matrix(17,-48,6,-17) (8/3,3/1) -> (8/3,3/1) Reflection Matrix(5,-16,1,-3) (3/1,4/1) -> (4/1,1/0) Parabolic IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL MULTI-ENDOMORPHISM FIXED POINT OF IMAGE This map is 2-valued. Matrix(1,0,0,-1) -> Matrix(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) -> Matrix(1,1,0,-1) -> (-1/2,1/0) Matrix(1,0,1,-1) -> Matrix(-1,0,2,1) (0/1,2/1) -> (-1/1,0/1) -> Matrix(1,1,0,-1) -> (-1/2,1/0) Matrix(7,-16,3,-7) -> Matrix(-1,0,2,1) (2/1,8/3) -> (-1/1,0/1) -> Matrix(1,1,0,-1) -> (-1/2,1/0) Matrix(17,-48,6,-17) -> Matrix(-1,0,2,1) (8/3,3/1) -> (-1/1,0/1) -> Matrix(1,1,0,-1) -> (-1/2,1/0) Matrix(5,-16,1,-3) -> Matrix(1,0,0,1) -> Matrix(1,1,-2,-1) (-1/1,0/1).(-1/2,1/0) Matrix(1,0,0,1) -> Matrix(1,0,0,1) -> Matrix(1,1,-2,-1) (-1/1,0/1).(-1/2,1/0)