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 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 -1/1 1/0 -3/1 -1/1 -2/1 -1/1 0/1 -3/2 -1/1 -4/3 -1/1 -1/2 -5/4 -1/2 -6/5 -2/3 0/1 -1/1 -1/2 0/1 0/1 1/1 1/2 4/3 1/2 1/1 3/2 1/1 2/1 0/1 1/1 3/1 1/1 4/1 1/1 1/0 5/1 1/0 6/1 0/1 2/1 1/0 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 IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(5,24,-4,-19) -> Matrix(1,2,-2,-3) Matrix(7,24,2,7) -> Matrix(1,2,0,1) Matrix(5,12,2,5) -> Matrix(1,0,2,1) Matrix(7,12,4,7) -> Matrix(1,0,2,1) Matrix(17,24,12,17) -> Matrix(3,2,4,3) Matrix(29,36,4,5) -> Matrix(1,0,2,1) Matrix(31,36,6,7) -> Matrix(1,0,2,1) Matrix(1,0,2,1) -> Matrix(1,0,4,1) Matrix(19,-24,4,-5) -> 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 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: ((2,4,3)(5,6,7); (1,4,7,8,5,2)(3,6); (1,2,6,8,7,3)(4,5)) ----------------------------------------------------------------------- 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 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 0/1 1/1 1/2 4/3 1/2 1/1 3/2 1/1 2/1 0/1 1/1 3/1 1/1 4/1 1/1 1/0 5/1 1/0 6/1 0/1 2/1 1/0 1/0 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 ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,2,1) Matrix(19,-24,4,-5) -> Matrix(3,-2,2,-1) Matrix(17,-24,5,-7) -> Matrix(3,-2,2,-1) Matrix(7,-12,3,-5) -> Matrix(1,0,0,1) Matrix(7,-36,1,-5) -> Matrix(1,0,0,1) 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 2 1 2/1 (0/1,1/1) 0 3 3/1 1/1 1 2 4/1 (1/1,1/0) 0 3 6/1 (1/1,1/0) 0 1 1/0 1/0 1 6 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 ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,1,-1) -> Matrix(1,0,2,-1) (0/1,2/1) -> (0/1,1/1) Matrix(5,-12,2,-5) -> Matrix(1,0,2,-1) (2/1,3/1) -> (0/1,1/1) Matrix(7,-24,2,-7) -> Matrix(-1,2,0,1) (3/1,4/1) -> (1/1,1/0) Matrix(5,-24,1,-5) -> Matrix(-1,2,0,1) (4/1,6/1) -> (1/1,1/0) Matrix(-1,12,0,1) -> Matrix(-1,2,0,1) (6/1,1/0) -> (1/1,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.