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 1/1 3/2 2/1 3/1 4/1 6/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,24,-4,-19) (-4/1,1/0) -> (-4/3,-5/4) Hyperbolic Matrix(7,24,2,7) (-4/1,-3/1) -> (3/1,4/1) Hyperbolic Matrix(5,12,2,5) (-3/1,-2/1) -> (2/1,3/1) Hyperbolic Matrix(7,12,4,7) (-2/1,-3/2) -> (3/2,2/1) Hyperbolic Matrix(17,24,12,17) (-3/2,-4/3) -> (4/3,3/2) 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(19,-24,4,-5) (1/1,4/3) -> (4/1,5/1) Hyperbolic 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 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 0/1 -1/2 1/0 1/1 -1/1 0/1 4/3 -1/2 1/0 3/2 -1/1 0/1 2/1 -1/2 1/0 3/1 -1/1 0/1 4/1 -1/2 1/0 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(19,-24,4,-5) (1/1,4/3) -> (4/1,5/1) Hyperbolic Matrix(17,-24,5,-7) (4/3,3/2) -> (3/1,4/1) Hyperbolic Matrix(7,-12,3,-5) (3/2,2/1) -> (2/1,3/1) 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(19,-24,4,-5) -> Matrix(1,1,-2,-1) -> Matrix(1,0,0,1) Matrix(17,-24,5,-7) -> Matrix(1,1,-2,-1) -> Matrix(1,0,0,1) Matrix(7,-12,3,-5) -> Matrix(1,0,0,1) -> Matrix(1,1,-2,-1) Matrix(7,-36,1,-5) -> 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 This is a reflection group. 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) 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(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 Matrix(5,-24,1,-5) (4/1,6/1) -> (4/1,6/1) Reflection Matrix(-1,12,0,1) (6/1,1/0) -> (6/1,1/0) Reflection 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(5,-12,2,-5) -> Matrix(1,1,0,-1) (2/1,3/1) -> (-1/2,1/0) -> Matrix(-1,0,2,1) -> (-1/1,0/1) Matrix(7,-24,2,-7) -> Matrix(-1,0,2,1) (3/1,4/1) -> (-1/1,0/1) -> Matrix(1,1,0,-1) -> (-1/2,1/0) Matrix(5,-24,1,-5) -> Matrix(-1,0,2,1) (4/1,6/1) -> (-1/1,0/1) -> Matrix(1,1,0,-1) -> (-1/2,1/0) Matrix(-1,12,0,1) -> Matrix(-1,0,2,1) (6/1,1/0) -> (-1/1,0/1) -> Matrix(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)