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): 72 Minimal number of generators: 13 Number of equivalence classes of cusps: 12 Genus: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -2/1 -1/1 -1/2 0/1 1/3 1/2 2/3 1/1 3/2 2/1 3/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,6,0,1) (-3/1,1/0) -> (3/1,1/0) Parabolic Matrix(7,18,12,31) (-3/1,-5/2) -> (1/2,3/5) Hyperbolic Matrix(5,12,12,29) (-5/2,-2/1) -> (2/5,1/2) Hyperbolic Matrix(7,12,18,31) (-2/1,-5/3) -> (1/3,2/5) Hyperbolic Matrix(19,30,12,19) (-5/3,-3/2) -> (3/2,5/3) Hyperbolic Matrix(5,6,-6,-7) (-3/2,-1/1) -> (-1/1,-3/4) Parabolic Matrix(25,18,18,13) (-3/4,-2/3) -> (4/3,3/2) Hyperbolic Matrix(19,12,30,19) (-2/3,-3/5) -> (3/5,2/3) Hyperbolic Matrix(31,18,12,7) (-3/5,-1/2) -> (5/2,3/1) Hyperbolic Matrix(29,12,12,5) (-1/2,-2/5) -> (2/1,5/2) 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(7,-6,6,-5) (2/3,1/1) -> (1/1,4/3) 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 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. ----------------------------------------------------------------------- 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): 9 Minimal number of generators: 4 Number of equivalence classes of elliptic points of order 2: 3 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 2 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 1/1 2/1 1/0 0/1 0/1 2/1 1/2 1/1 2/1 1/0 1/1 1/1 2/1 1/0 1/0 1/1 2/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-1,-1) (-1/1,1/0) -> (-1/1,1/0) Elliptic Matrix(1,0,3,1) (-1/1,0/1) -> (0/1,1/2) Parabolic Matrix(3,-2,5,-3) (1/2,1/1) -> (1/2,1/1) Elliptic Matrix(1,-2,1,-1) (1/1,1/0) -> (1/1,1/0) Elliptic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-1,-1) -> Matrix(1,0,0,1) Matrix(1,0,3,1) -> Matrix(1,0,0,1) Matrix(3,-2,5,-3) -> Matrix(1,0,0,1) Matrix(1,-2,1,-1) -> Matrix(1,0,0,1) ----------------------------------------------------------------------- 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 (0/1,2/1) 1/1 (0/1,2/1) 1/0 (0/1,2/1) GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,-1) (-1/1,1/0) -> (-1/1,1/0) Reflection Matrix(-1,0,1,1) (-2/1,0/1) -> (-2/1,0/1) Reflection Matrix(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(1,-2,1,-1) (1/1,1/0) -> (1/1,1/0) Elliptic IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(1,0,1,-1) (-1/1,1/0) -> (0/1,2/1) Matrix(-1,0,1,1) -> Matrix(1,0,1,-1) (-2/1,0/1) -> (0/1,2/1) Matrix(1,0,2,-1) -> Matrix(1,0,1,-1) (0/1,1/1) -> (0/1,2/1) Matrix(1,-2,1,-1) -> Matrix(1,0,0,1)