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 -3/1 -12/5 -3/2 0/1 1/1 3/2 9/5 2/1 3/1 9/2 6/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,18,-2,-7) (-3/1,1/0) -> (-3/1,-5/2) Parabolic Matrix(59,144,34,83) (-5/2,-12/5) -> (12/7,7/4) Hyperbolic Matrix(61,144,36,85) (-12/5,-7/3) -> (5/3,12/7) Hyperbolic Matrix(47,108,10,23) (-7/3,-9/4) -> (9/2,5/1) Hyperbolic Matrix(25,54,6,13) (-9/4,-2/1) -> (4/1,9/2) Hyperbolic Matrix(11,18,-8,-13) (-2/1,-3/2) -> (-3/2,-4/3) Parabolic Matrix(41,54,22,29) (-4/3,-9/7) -> (9/5,2/1) Hyperbolic Matrix(85,108,48,61) (-9/7,-5/4) -> (7/4,9/5) Hyperbolic Matrix(29,36,4,5) (-5/4,-6/5) -> (6/1,1/0) Hyperbolic Matrix(31,36,6,7) (-6/5,-1/1) -> (5/1,6/1) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(13,-18,8,-11) (1/1,3/2) -> (3/2,5/3) Parabolic Matrix(7,-18,2,-5) (2/1,3/1) -> (3/1,4/1) 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): 36 Minimal number of generators: 7 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: 8 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 3/2 12/7 2/1 3/1 9/2 6/1 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 3/2 -1/1 0/1 5/3 -1/1 0/1 12/7 -1/2 1/0 7/4 -1/1 0/1 9/5 -1/1 0/1 2/1 -1/2 1/0 3/1 -1/1 0/1 4/1 -1/2 1/0 9/2 -1/1 0/1 5/1 -1/1 0/1 6/1 -1/2 1/0 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,-18,8,-11) (1/1,3/2) -> (3/2,5/3) Parabolic Matrix(85,-144,49,-83) (5/3,12/7) -> (12/7,7/4) Parabolic Matrix(61,-108,13,-23) (7/4,9/5) -> (9/2,5/1) Hyperbolic Matrix(29,-54,7,-13) (9/5,2/1) -> (4/1,9/2) Hyperbolic Matrix(7,-18,2,-5) (2/1,3/1) -> (3/1,4/1) Parabolic Matrix(7,-36,1,-5) (5/1,6/1) -> (6/1,1/0) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL MULTI-ENDOMORPHISM This map is 2-valued. Matrix(1,0,1,1) -> Matrix(1,1,-2,-1) -> Matrix(1,0,0,1) Matrix(13,-18,8,-11) -> Matrix(1,0,0,1) -> Matrix(1,1,-2,-1) Matrix(85,-144,49,-83) -> Matrix(1,1,-2,-1) -> Matrix(1,0,0,1) Matrix(61,-108,13,-23) -> Matrix(1,1,-2,-1) -> Matrix(1,0,0,1) Matrix(29,-54,7,-13) -> Matrix(1,1,-2,-1) -> Matrix(1,0,0,1) Matrix(7,-18,2,-5) -> Matrix(1,0,0,1) -> Matrix(1,1,-2,-1) Matrix(7,-36,1,-5) -> Matrix(1,1,-2,-1) -> Matrix(1,0,0,1) 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) 3/1 (-1/1,0/1).(-1/2,1/0) 4/1 (-1/1,0/1).(-1/2,1/0) 9/2 (-1/1,0/1).(-1/2,1/0) 5/1 (-1/1,0/1).(-1/2,1/0) 6/1 (-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,-18,2,-5) (2/1,3/1) -> (3/1,4/1) Parabolic Matrix(17,-72,4,-17) (4/1,9/2) -> (4/1,9/2) Reflection Matrix(19,-90,4,-19) (9/2,5/1) -> (9/2,5/1) Reflection Matrix(7,-36,1,-5) (5/1,6/1) -> (6/1,1/0) 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(1,1,0,-1) (0/1,2/1) -> (-1/2,1/0) -> Matrix(-1,0,2,1) -> (-1/1,0/1) Matrix(7,-18,2,-5) -> Matrix(1,0,0,1) -> Matrix(1,1,-2,-1) (-1/1,0/1).(-1/2,1/0) Matrix(17,-72,4,-17) -> Matrix(-1,0,2,1) (4/1,9/2) -> (-1/1,0/1) -> Matrix(1,1,0,-1) -> (-1/2,1/0) Matrix(19,-90,4,-19) -> Matrix(-1,0,2,1) (9/2,5/1) -> (-1/1,0/1) -> Matrix(1,1,0,-1) -> (-1/2,1/0) Matrix(7,-36,1,-5) -> Matrix(1,1,-2,-1) (-1/1,0/1).(-1/2,1/0) -> Matrix(1,0,0,1) Matrix(1,0,0,1) -> Matrix(1,0,0,1) -> Matrix(1,1,-2,-1) (-1/1,0/1).(-1/2,1/0)