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 -1/6 0/1 1/4 1/3 1/2 2/3 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 1/1 -3/4 1/1 1/0 -2/3 1/0 -1/2 0/1 -1/3 1/2 -1/4 1/2 1/1 -1/5 2/3 1/1 -1/6 1/1 0/1 0/1 1/1 1/4 1/1 1/0 1/3 1/0 1/2 0/1 2/3 1/2 3/4 1/2 1/1 4/5 2/3 1/1 5/6 1/1 1/1 0/1 1/1 1/0 1/2 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(5,4,-24,-19) (-1/1,-3/4) -> (-1/4,-1/5) Hyperbolic Matrix(17,12,24,17) (-3/4,-2/3) -> (2/3,3/4) Hyperbolic Matrix(7,4,12,7) (-2/3,-1/2) -> (1/2,2/3) Hyperbolic Matrix(5,2,12,5) (-1/2,-1/3) -> (1/3,1/2) Hyperbolic Matrix(7,2,24,7) (-1/3,-1/4) -> (1/4,1/3) Hyperbolic Matrix(31,6,36,7) (-1/5,-1/6) -> (5/6,1/1) Hyperbolic Matrix(29,4,36,5) (-1/6,0/1) -> (4/5,5/6) Hyperbolic Matrix(19,-4,24,-5) (0/1,1/4) -> (3/4,4/5) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,0,1) Matrix(5,4,-24,-19) -> Matrix(1,-2,2,-3) Matrix(17,12,24,17) -> Matrix(1,-2,2,-3) Matrix(7,4,12,7) -> Matrix(1,0,2,1) Matrix(5,2,12,5) -> Matrix(1,0,-2,1) Matrix(7,2,24,7) -> Matrix(3,-2,2,-1) Matrix(31,6,36,7) -> Matrix(3,-2,2,-1) Matrix(29,4,36,5) -> Matrix(1,-2,2,-3) Matrix(19,-4,24,-5) -> Matrix(1,-2,2,-3) 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): 2 Degree of the the map X: 2 Degree of the the map Y: 8 Permutation triple for Y: ((1,4,7,8,5,2)(3,6); (1,2,6,8,7,3)(4,5); (2,4,3)(5,6,7)) ----------------------------------------------------------------------- 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): 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 1/4 1/3 1/2 5/6 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/4 1/1 1/0 1/3 1/0 1/2 0/1 2/3 1/2 3/4 1/2 1/1 4/5 2/3 1/1 5/6 1/1 1/1 0/1 1/1 1/0 1/2 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,1,0,1) (0/1,1/0) -> (1/1,1/0) Parabolic Matrix(19,-4,24,-5) (0/1,1/4) -> (3/4,4/5) Hyperbolic Matrix(17,-5,24,-7) (1/4,1/3) -> (2/3,3/4) Hyperbolic Matrix(7,-3,12,-5) (1/3,1/2) -> (1/2,2/3) Parabolic Matrix(31,-25,36,-29) (4/5,5/6) -> (5/6,1/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,1,0,1) -> Matrix(1,0,0,1) Matrix(19,-4,24,-5) -> Matrix(1,-2,2,-3) Matrix(17,-5,24,-7) -> Matrix(1,-2,2,-3) Matrix(7,-3,12,-5) -> Matrix(1,0,2,1) Matrix(31,-25,36,-29) -> Matrix(3,-2,2,-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 elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 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 c d 0/1 (0/1,1/1) 0 6 1/6 1/1 2 1 1/4 (1/1,1/0) 0 3 1/3 1/0 1 2 1/2 0/1 2 3 1/0 (0/1,1/1) 0 1 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(1,0,12,-1) (0/1,1/6) -> (0/1,1/6) Reflection Matrix(5,-1,24,-5) (1/6,1/4) -> (1/6,1/4) Reflection Matrix(7,-2,24,-7) (1/4,1/3) -> (1/4,1/3) Reflection Matrix(5,-2,12,-5) (1/3,1/2) -> (1/3,1/2) Reflection Matrix(-1,1,0,1) (1/2,1/0) -> (1/2,1/0) 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) -> (0/1,1/1) Matrix(1,0,12,-1) -> Matrix(1,0,2,-1) (0/1,1/6) -> (0/1,1/1) Matrix(5,-1,24,-5) -> Matrix(-1,2,0,1) (1/6,1/4) -> (1/1,1/0) Matrix(7,-2,24,-7) -> Matrix(-1,2,0,1) (1/4,1/3) -> (1/1,1/0) Matrix(5,-2,12,-5) -> Matrix(1,0,0,-1) (1/3,1/2) -> (0/1,1/0) Matrix(-1,1,0,1) -> Matrix(1,0,2,-1) (1/2,1/0) -> (0/1,1/1) ----------------------------------------------------------------------- The pullback map has no extra symmetries.